Applications of Harmonic Balance Method in Periodic Flows Gregor Cvijetić and Inno Gatin Faculty of Mechanical Engineering and Naval Architecture, Zagreb 10000, Croatia, gregor.cvijetic@gmail.com The Harmonic Balance method for periodic flows is presented in this work. The Harmonic Balance method deploys Fourier series expansion based on the assumption that variables are periodic and can be accurately represented by a mean value and first n harmonics. This procedure yields a number of coupled steady state equations, replacing the single transient equation. The number of equations depends on the number of harmonics chosen: for n harmonics +1 equations are solved. The coupling between equations is achieved through source terms which account for the time derivative term in transient equation. This paper deals with two distinct applications of the Harmonic Balance method: turbomachinery and waves on free surface. The turbomachinery test case presented is ERCOFTAC centrifugal pump consisting of rotor and stator. Due to presence of different frequencies in rotor and stator, the multiple frequency approach is used. Harmonic Balance method is compared against conventional transient solver, comparing efficiency, head and torque in several time instants. Pressure contours on rotor blade are compared to present the local accuracy of the method. The second test case deals with regular surface wave propagation with current. For the mildly steep wave imposed, the validation is performed comparing the surface elevation at the middle of domain. The Harmonic Balance surface elevation and transient surface elevation are compared in frequency domain using mean value and first harmonic. The method is implemented in foam-extend, a community driven fork of the open source CFD software, OpenFOAM. INTRODUCTION Many modern problems include different periodic flows such as wave-like phenomena, wing oscillations, moving valves and similar problems with either periodic boundary conditions or periodic body motions. In most cases transient simulation is used for such problems as transient flow features are wanted. Usually a number of periods have to be run in order to obtain periodic steady state and neglect simulation start unsteadiness, thus yielding long CPU time. The need for reducing CPU time, but still preserving periodic flow features motivated the development of new methods. Harmonic Balance [1,2] is a quasi-steady state method developed for solving periodic flows and problems with prescribed harmonic motion. Deploying the Fourier expansion, Harmonic Balance method is capable of resolving transient flow features while solving a set of steady state equations. Modelling the transient transport equation in form of Harmonic Balance formulates a set of steady state equations that are mutually coupled. The coupling term replaces the time derivative term in transient equations. Such approach makes Harmonic Balance efficient and accurate as it is capable of resolving transient flow features while offering a significant CPU time reduction [3]. The Harmonic Balance method was initially developed as a periodic boundary condition by He [4] and later extended for solving the two-dimensional Navier-Stokes equations by He and Ning [5]. The Harmonic Balance method is undergoing extensive development in wide area of applications.
Recently, Dufour et al. [6] have presented oscillating airfoils and Thomas et al. [7] have presented oscillating wings. Hall et al. [11] used complex geometries to extend and demonstrate the Harmonic Balance capabilities, which was later used by Nadarajah et al. [12] in performing shape design optimization. After development of Harmonic Balance for turbomachinery, the need for multiple frequencies was stressed by Gopinath et al. [13]. In case of multistage turbines rotor frequency changes in each stage due to a different number of blades. A similar multiple frequency by Guedeney et al. [14] relies on a uniform time sampling of the longest period of interest, therefore reconstructing the flow field between stages to match the time instants. Additionally, Guedeney et al. [15] presented the two algorithms for non-uniform time sampling in multiple frequency approach of Harmonic Balance. We present the Harmonic Balance method for two distinct problems: turbomachinery and wave propagation. Turbomachinery test case is a single phase problem which employs multiple frequency approach to account for rotor-stator interaction. Validation is performed using the ERCOFTAC centrifugal pump test case consisting of rotor and stator. Comparison of pressure contours on rotor blade, local velocity field and global pump parameters against the transient simulation is presented. Regular wave propagation test case is a two phase problem with single frequency. The wave imposed is a mildly steep wave and convergence with the increasing number of harmonics is presented. In order to eliminate the wave reflection from the boundaries, relaxation zones are used at the inlet and outlet of the domain. MATHEMATICAL MODEL This section presents the mathematical model of the Harmonic Balance (HB) method. HB treatment, transforming the time-derivative term into a source term, is done on scalar transport equation and extended to incompressible flow equations. Mathematical model presented here is general and valid for any number of harmonics. Scalar transport equation can be written in the following form: Q R 0, t where R stands for convection and diffusion transport and source/sink terms: R ( u Q) ( Q) S, (2) Introducing the Fourier series expansion of variable Q with n harmonics reads: Q( t) Q Q sin ( i t) Q cos( i t). 0 n i Si Ci Q (3) Scripture characters, Q, denote the time domain variables, while Q denotes frequency domain field. Fourier expansion of R is analogous to the one in Eqn. (3), with Q substituted with R. Inserting the Eqn. (3) in Eqn. (1) yields sine and cosine terms, which after grouping give +1 equations: n for sine terms, n for cosine and 1 for mean value. Therefore, HB formulation of scalar transport equations becomes a set of +1 equations, written in matrix form: (1) AQ R 0, (4) where A is a () ( 1) coefficients matrix, Q and R are column matrices containing Fourier sine and cosine coefficients and ω is a base radian frequency.
In order to easily switch between the time and frequency domain, a matrix representation of Direct Fourier Transform (DFT) is introduced: Q EQ, Q=E Q, (5) (6) where Q is a column vector of solution in +1 time instants: where t i stands for: T Q Q Q... Q t, 1 t2 t (7) 1 it ti, for i.... With the aid of matrices E i E and multiplying the Eqn. (4) from the left, scalar transport equation in time domain is obtained. Time derivative termi s replaced with a source term: AEQ ER 0. (9) Equation (9) represents a set of +1 coupled steady state equations. Expanded form of HB scalar transport equation can be written: 2 ( u Qt ) ( Q ) S, za 1...2 1, j t j Qt P j i jqt j n (10) j where P i is: n k i Pi k sin( ik t), for i 0..., (11) T t. (8) (12) For solving incompressible single phase problems, continuity equation and momentum equation are obtained using velocity vector u instead of scalar variable: u 0, (13) u p ( uu) ( u), (14) t where is kinematic viscosity, ρ is fluid density and p is pressure. The momentum equation derivation is analogous to the scalar transport equation derivation. Continuity equation remains unchanged as there is no time derivative term: u 0. (15) t j Momentum equation in form of Harmonic Balance states: 2 ( ut u ) ( ), for 1... 2 1. j t u j t p j t P j i ju t j n (16) i i
RESULTS The Harmonic Balance (HB) method is validated for two applications: turbomachinery and naval hydrodynamics. Turbomachinery test case is a single phase turbulent problem presented on ERCOFTAC centrifugal pump geometry. In order to accurately model rotor stator interaction, the multiple frequency approach has to be used along with HB. This ensures the flow field is solved for correct frequencies separately in rotor and in stator. The comparison of global pump parameters as well as local flow transients is presented. Naval hydrodynamics test case is a two phase flow problem with mildly steep wave imposed. ERCOFTAC Centrifugal Pump A 2-D test case consisting of rotor and stator was chosen to present the HB capabilities for turbomachinery applications. Radial inlet velocity is prescribed, u=11.4 m/s with rotational speed of 2000 rpm and for k turbulence modeling the model is used. It should be noted that different base frequencies appear in rotor and stator, meaning that single frequency HB method would not yield accurate results. For such purposes a multiple frequency approach is adopted. Multiple frequency approach separates the domain into two parts depending on frequency: rotor frequency and stator frequency, thus developing the initial Fourier series expansion with different frequencies in each zone. The frequencies used are: f f rotor stator rpm / 60, f rotor n. rotor blades (17) This shows that the base frequency affecting the rotor is the number of revolutions per second, while the base frequency in stator is the number of a single rotor blade passing by the stator blade per second. Table 1 presents the global pump parameters comparison. Harmonic Balance results are compared to conventional transient simulation and steady state simulation. MRF is a multiple reference frame type of steady state simulation where rotation is taken into account by adding additional source terms. Harmonic Balance simulation was run using 1 and 2 harmonics. The comparison of efficiency, head and torque in three different time instants is given. The largest error for the HB simulation with 1 harmonic is 2%, while in case of 2 harmonics it is 1.3%, compared to conventional transient simulation. On the other hand, the error of steady state simulation results goes up to 3.6%. Table 1. Global pump parameters comparison t1 t2 t3 Transient MRF Error, % HB, 1h Error, % HB, 2h Error, % Efficiency 89,72 89,65 0.01 88.80 1.0 89.76 0.0 Head 81,48 84,12 3.2 81.80 0.4 80.45 1.3 Torque 0,0297 0,0307 3.3 0.0302 1.7 0.0294 0.9 Efficiency 89,92 89,65 0.3 88.78 1.3 89.81 0.1 Head 81,48 84,12 3.2 81.85 0.4 80.60 1.1 Torque 0,0296 0,0307 3.6 0,0302 2.0 0.0295 0.4 Efficiency 89,83 89,65 0.2 88.85 1.1 89.71 0.1 Head 81,49 84,12 3.2 81.79 0.4 80.39 1.3 Torque 0,0297 0,0307 3.4 0,0302 1.6 0.0294 1.0
Local transient flow features and the ability of the method to capture it is an important aspect in transient problems where such instabilities occur. Figure 1 presents the comparison of local flow field in stator blade passages. In figure 1a wakes in the stator blade passage can be noticed, while in the case of 1 harmonic wakes exist but are not as noticeable. In case of 2 harmonics wakes are resolved more accurately, but still not as accurate as in transient simulation. In figure 1d no wakes exist as the simulation is steady state and the rotor position is fixed. a) Transient simulation, b) HB with 1 harmonic, c) HB with 2 harmonics, d) MRF simulation. Figure 1. Local flow field transients comparison. Two phase propagating surface wave A viscid two phase simulation of regular wave propagation with mean current is chosen to present the HB capabilities for naval hydrodynamics applications. The wave imposed is mildly steep with T=4as and H=0.3 m, with the current of 4 m/s, resulting in wave length of 43.6 m and wave steepness ka=0.0216. In order to use sufficiently long relaxation zones at the inlet and outlet, the length of the domain is set to be 174.4 m. Figure 2 presents the initial condition for t=t, where the magnitude of the initial velocity field is shown and the position of the free surface is clearly visible. Figure 2. Initial condition in t=t. In table 2 the results obtained using 1, 2, 3, 4, 5, 6, 7 and 8 resolved harmonics are presented. Iterative uncertainties and differences with the transient solution are given, where the differences of the 0th
order are given as absolute due to very small values. For the 0th order the iterative uncertainties are below 1%, while for the 1st order the iterative uncertainties are negligible. No. harmonics a,0, m Table 2. Comparison of HB results and transient simulation a,1, m U I,0, % U I,1, % 0, m 1, % 1 0.001898 0.1531 0.5962 0.03494-0.0006-0.328 2 0.000302 0.1520 0.9360 0.01349 0.0010 0.393 3 0.000411 0.1519 0.3916 0.01975 0.0009 0.459 4 0.000394 0.1518 0.1845 0.00099 0.0009 0.524 5 0.000352 0.1517 0.1215 0.00033 0.0009 0.590 6 0.000438 0.1517 0.1651 0.00033 0.0009 0.590 7 0.000337 0.1516 0.1386 0.00033 0.0010 0.655 8 0.000332 0.1516 0.1008 0.00033 0.0010 0.655 It can be noticed in figure 3 that with the increasing number of resolved harmonics the iterative uncertainty reduces. Comparing the HB simulation with the conventional transient simulation, the errors are below 1% for the first order in all cases. Furthermore, it can be noticed that the higher order harmonic amplitudes have minor influence compared to first order, as the differences for the first order do not decrease with the increasing number of harmonics. The presented behavior shows that for a mildly steep wave only one harmonic is sufficient. Figure 3. Wave propagation convergence. CONCLUSION This paper presents the Harmonic Balance method for non-linear single phase and two phase periodic flows. The method is validated using two distinct test cases: turbomachinery test case and a naval hydrodynamics test case. Both test cases present the accuracy of the presented method and its applicability to a wide variety of problems. The turbomachinery results show good agreement with
transient simulation and the transient flow features that the method is able to resolve even in cases of small number of harmonics. Regular wave propagation test case has demonstrated that the HB method can also be used in two phase problems as well as in problems related to naval hydrodynamics. It was shown that one harmonic is sufficient for simulating regular mildly steep wave where the errors of the compared features are below 1%. In the final paper also the efficiency in terms of CPU time will be presented, as well as the speed-up achieved compared to conventional transient simulation. References: [1] Hall, K.C., Thomas, J.P., Clark, W.S., Computation of unsteady nonlinear flows in cascades using a harmonic balance technique, AIAA Journal, Vol. 50., No. 5., 2002., str. 879-886 [2] Chen, T., P. Vasanthakumar, and L. He. Analysis of unsteady blade row interaction using nonlinear harmonic approach. ASME Turbo Expo 2000: Power for Land, Sea, and Air. American Society of Mechanical Engineers, 2000. [3] Cvijetić, G., Jasak, H., Vukčević, V., Finite volume implementation of harmonic balance method for periodic non-linear flows, 54th AIAA Aerospace Sciences Meeting, 2016. [4] He, L., Method of simulating unsteady turbomachinery flows with multiple perturbations, AIAA Journal, Vol. 30., No. 11., 1992., str. 2730-2735. [5] He, L., Ning, W., Efficient approach for analysis of unsteady viscous flows in turbomachines, AIAA Journal, Vol. 36., No. 11., 1998., str. 2005-2012. [6] Dufour, G., Sicot, F., Puigt, G., Liauzun, C., Contrasting the harmonic balance and linearized methods for oscillating-flap simulations, AIAA Journal, Vol. 48, No. 4., 2010. [7] Thomas, J.P., Custer, C.H., Dowell, E.H., Hall, K.C., Unsteady flow computation using a harmonic balance approach implemented about the OVERFLOW 2 flow solver, 19 th AIAA Computational Fluid Dynamics Conference, 2009. [8] Ekici, Kivanc, and Kenneth C. Hall. Harmonic balance analysis of limit cycle oscillations in turbomachinery. AIAA journal 49.7 (2011): 1478-1487. [9] Sicot, Frédéric, et al. Time-Domain Harmonic Balance Method for Turbomachinery Aeroelasticity. AIAA journal 52.1 (2013): 62-71. [10] Ekici, K., Thomas, J.P., Hall, K.C., Voytovych, D.M., Frequency domain techniques for complex and non-linear flows in turbomachinery, 33 rd AIAA Fluid Dynamics Conference and Exhibit, 2003. [11] Nadarajah, Siva, and Antony Jameson. Optimum shape design for unsteady threedimensional viscous flows using a nonlinear frequency-domain method. Journal of Aircraft 44.5 (2007): 1513-1527.
[12] Gopinath, A., Weide, E., Alonso, J.J., Jameson, A., Ekici, K., Hall, K.C., Three-dimensional unsteady multi-stage turbomachinery simulations using the harmonic balance technique, Collection of Technical Papers 45 th AIAA Aerospace Sciences Meeting, 2007. [13] Guédeney, T., Gomar, A., Sicot, F., Multi-frequential harmonic balance approach for the computation of unsteadiness in multi-stage turbomachines, 21ème Congrès Français de Mécanique, Bordeaux, 2013. [14] Guédeney, Thomas, et al. Non-uniform time sampling for multiple-frequency harmonic balance computations. Journal of Computational Physics 236 (2013): 317-345.