FSBHarmonic Balance Method
for Turbomachinery ApplicationsGregor Cvijetic and Hrvoje Jasak
[email protected] [email protected]
AbstractThis work presents the Harmonic Balance method for in-compressible turbulent periodic flows. The method isimplemented and tested in foam-extend, a community–driven fork of the open source software OpenFOAM. Themethod is validated using ERCOFTAC centrifugal pumptest case. Results are presented and compared with con-ventional transient simulation. Local transient effects andglobal pump parameters are compared.
Mathematical ModelPrimitive variables are expressed by a Fourier series in time, with n harmonics. Substi-tuting the variables in transport equations with Fourier series, 2n+1 coupled equationsare obtained:• Harmonic Balance continuity equation: ∇•ut j = 0,
• Harmonic Balance momentum equation: ∇•(ut j ut j )−∇•(γ∇ut j ) =−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 spacedtime steps within a period are obtained. Each of the 2n+ 1 equations represents onetime instant. Equations without the time derivative term in its original form (continuityequation) remain the same, using variables corresponding to the time instant currentlycalculated.
IntroductionThe Harmonic Balance method is a quasi-steady statemethod developed for simulating non-linear temporallyperiodic flows. It is based on the assumption thateach primitive variable can be accurately presented bya Fourier series in time, using first n harmonics and themean value. Such assumption allows replacing the timederivative term in transport equations with coupled sourceterms, thus transforming the transient equations into acoupled set of quasi steady equations. The benefit com-pared to steady state methods is that Harmonic Balanceis able to describe the transient effects of periodic flows,while providing significant speed-up compared to tran-sient simulation.
ERCOFTAC centrifugal pumpERCOFTAC Centrifugal Pump is simulated using Harmonic Balance method and com-pared against conventional transient solver. The geometry is a 2D simplified model of acentrifugal turbomachine, discretised with 93 886 hexahedral cells. The pump consistsof rotor (inner) and stator (outer) part. The rotation speed is 2000 rpm with inlet veloc-ity set to 11.4 m/s and k-Epsilon model used for turbulence. Additionally, in HarmonicBalance simulations multiple frequency approach was used to account for different fre-quencies in the stator and rotor.
-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 rotorblade surface at 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 rotorblade surface at 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 rotorblade surface at t = T .
Harmonic Balance simulations were run using 1 and 2 harmonics, comparing the effi-ciency, head and torque, and pressure on the rotor blade surface with transient simulationresults. Figure 1 shows the comparison of pressure on the rotor blade surface for thetime instant t = T/3. Figures 2 and 3 show the comparison at time instants t = 2T/3 andt = T , respectively. The results for 2 harmonics agree better with the transient solutionthen in case of 1 harmonic.
Figure 4: Transient solution. Figure 5: Harmonic Balancewith 1 harmonic.
Figure 6: Harmonic Balancewith 2 harmonics.
Local transient effects are presented in Figures 4-6. Figure 4 presents the rotor wakes instator blade passage. Wakes can be noticed in the Harmonic Balance solution, Figures 5and 6, but are not resolved as accurate due to small number of harmonics used.
Global Pump ParametersTable 1: Pump characteristics comparison
Transient HB, 1h error, % HB, 2h error, %Efficiency 89.72 88.80 1.0 89.76 0.0
t = T3 Head 81.48 81.80 0.4 80.45 1.3
Torque 0.0297 0.0302 1.7 0.0294 0.9
Efficiency 89.92 88.78 1.3 89.81 0.1t = 2T
3 Head 81.48 81.85 0.4 80.6 1.1Torque 0.0296 0.0302 2.0 0.0295 0.4
Efficiency 89.83 88.85 1.1 89.71 0.1t = T Head 81.49 81.79 0.4 80.39 1.3
Torque 0.0297 0.0302 1.6 0.0294 1.0
Table 1 presents a comparison of pump characteristicsobtained using the Harmonic Balance and a transientsolver. Results are presented for efficiency, head andtorque at three time instants. In the compared featuresand time instants the error is lower than 2%, showing thecapability and accuracy of the Harmonic Balancemethod.
CPU Time ComparisonThe simulations were run in parallel using four cores onan Intel Core i5-3570K, 3.4 GHz computer. The signifi-cant CPU time reduction from transient to Harmonic Bal-ance simulation can be noticed: one period of transientsimulation took ∼5 hours of CPU time, while HarmonicBalance simulation with 1 harmonic took ∼52 minutesand nearly 3000 iterations. The 2 harmonics HarmonicBalance simulation took ∼78 minutes of CPU time, con-verging in approximately 2400 iterations. In transientruns, a number of periods have to be run before reach-ing fully periodic steady state. Assuming 10 periods areneeded to reach periodic steady state, full transient simu-lation takes 50 hours of CPU time, being 40 times slowerthan HB with 2 harmonics.
ConclusionThe Harmonic Balance method is presented for unsteady periodic non–linear flows inturbomachinery applications. The comparison of pressure contours around the rotorblade shows that the Harmonic Balance method is capable of capturing the transient flowfield accurately even in multi-frequential environment. Additional comparison of pumpcharacteristics with highest error being 2% shows that the Harmonic Balance methodcan be used as a part of a design process with accurate flow predictions and significantCPU time savings.
