Harmonic Balance Method forTurbomachinery Applications
Gregor Cvijetic1, Hrvoje Jasak1,2
1Faculty of Mechanical Engineering and Naval Architecture, Croatia2Wikki Ltd, United Kingdom
[email protected], [email protected] [email protected]
Abstract
This work presents the Harmonic Balance methodfor incompressible turbulent periodic flows. Themethod is implemented and tested in foam-extend, a community–driven fork of the opensource software OpenFOAM. One stage centrifu-gal pump is simulated using the Harmonic Bal-ance method with 1 and 2 harmonics. Results arepresented and compared with conventional tran-sient simulation.
1. Introduction
The Harmonic Balance method is a quasi-steadystate method developed for simulating non-lineartemporally periodic flows. It is based on the as-sumption that each primitive variable can be ac-curately presented by a Fourier series in time, us-ing first n harmonics and the mean value. Suchassumption allows replacing the time derivativeterm in transport equations with coupled sourceterms, thus transforming the transient equationsinto a coupled set of quasi steady equations. Thebenefit compared to steady state methods is thatHarmonic Balance is able to describe the transienteffects of periodic flows, while providing significantspeed-up compared to transient simulation.
2. Mathematical Model
Primitive variables are expressed by a Fourier se-ries in time, with n harmonics. Substituting thevariables in transport equations with Fourier se-ries, 2n + 1 coupled equations are obtained:
•Harmonic Balance continuity equation:
∇•utj = 0,
•Harmonic Balance momentum equation:
∇•(utjutj)−∇•(γ∇utj) = − 2ω
2n + 1
2n∑i=1
P(i−j)uti
,
where
Pi =
n∑k=1
k sin(kωi∆t), for i = {1,2n}.
Corresponding to the Fourier series expansion,for n harmonics 2n + 1 equally spaced time stepswithin a period are obtained. Each of the 2n + 1equations represents one time instant. Equationswithout the time derivative term in its original form(continuity equation) remain the same, using vari-ables corresponding to the time instant currentlycalculated.
3. Numerical Procedure
Second order accurate, polyhedral Finite VolumeMethod is used. Segregated solution algorithmSIMPLE is adopted. Each of the 2n+ 1 time stepsis resolved in its own SIMPLE loop.
4. ERCOFTAC Centrifugal Pump
ERCOFTAC Centrifugal Pump is simulated usingHarmonic Balance method and compared againstconventional transient solver. The geometry is a2D simplified model of a centrifugal turbomachine,discretised with 93 886 hexahedral cells. Thepump consists of rotor (inner) and stator (outer)part. The rotation speed is 2000 rpm with in-let velocity set to 11.4 m/s and k-Epsilon modelused for turbulence. Additionally, in HarmonicBalance simulations multiple frequency approachwas used to account for different frequencies inthe stator and rotor.Harmonic Balance simulations were run using 1and 2 harmonics, comparing the efficiency, headand torque, and pressure on the rotor blade sur-face with transient simulation results. Figure 1shows the comparison of pressure on the rotorblade surface for the time instant t = T/3. Fig-ures 2 and 3 show the comparison at time instantst = 2T/3 and t = T , respectively. The results for 2harmonics agree better with the transient solutionthen in case of 1 harmonic.
-0.2 -0.15 -0.1 -0.05 0
x-Axis
-1200
-1000
-800
-600
-400
-200
0
Pre
ssure
, P
a
TransientHB, 1h
HB, 2h
Figure 1: Pressure on rotor blade surfaceat t = T/3.
-0.15 -0.1 -0.05 0 0.05
x-Axis
-1200
-1000
-800
-600
-400
-200
0
Pre
ssu
re,
Pa
TransientHB, 1h
HB, 2h
Figure 2: Pressure on rotor blade surfaceat t = 2T/3.
-0.15 -0.1 -0.05 0 0.05
x-Axis
-1200
-1000
-800
-600
-400
-200
0
Pre
ssu
re,
Pa
TransientHB, 1h
HB, 2h
Figure 3: Pressure on rotor blade surfaceat t = T .
Table 1 presents a comparison of pump character-istics obtained using the Harmonic Balance anda transient solver. Results are presented for effi-ciency, head and torque at three time instants. Inthe compared features and time instants the er-ror is lower than 5%, showing the capability andaccuracy of the Harmonic Balance method.
Table 1: Pump characteristics comparisonTransient solver HB, 1h error, % HB, 2h error, %
Efficiency 89.72 93.55 4.26 90.07 0.39t = T
3 Head 81.48 83.37 2.32 83.04 1.92Torque 0.0297 0.0305 2.65 0.0303 2.03
Efficiency 89.92 92.07 2.38 93.85 4.36t = 2T
3 Head 81.48 83.45 2.41 83.13 2.02Torque 0.0296 0.0304 2.64 0.0303 2.28
Efficiency 89.83 89.63 0.22 91.68 2.07t = T Head 81.49 83.09 1.96 82.94 1.77
Torque 0.0297 0.0304 2.65 0.0303 2.28
ERCOFTAC centrifugal pump geometry and ve-locity field is presented in Figure 4. Figures a) andb) present the Harmonic Balance solutions for 1and 2 harmonics, respectively. Figure c) presentsthe transient solution velocity field.
Figure 4: a) Harmonic Balance with 1 harmonic;b) Harmonic Balance with 2 harmonics; c) tran-sient solver flow field at t = T .
5. Performance and hardware
The simulations were run in parallel using fourcores on an Intel Core i5-3570K, 3.4 GHz com-puter. The significant CPU time reduction fromtransient to Harmonic Balance simulation can benoticed: one period of transient simulation took∼5 hours of CPU time, while Harmonic Balancesimulation with 1 harmonic took ∼52 minutes andnearly 3000 iterations. The 2 harmonics Har-monic Balance simulation took ∼78 minutes ofCPU time, converging in approximately 2400 iter-ations. In transient runs, a number of periods haveto be run before reaching fully periodic steadystate. Thus, CPU time of 1 period should be mul-tiplied.
6. Conclusion
The Harmonic Balance method is presented forunsteady periodic non–linear flows in turboma-chinery applications. The comparison of pressurecontours around the rotor blade shows that theHarmonic Balance method is capable of captur-ing the transient flow field accurately even in multi-frequential environment. Additional comparisonof pump characteristics with highest error being4.36% shows that the Harmonic Balance methodcan be used as a part of a design process withaccurate flow predictions and significant CPU timesavings.
