Harmonic Balance Method forUnsteady Periodic Flows
Gregor Cvijetić1, Hrvoje Jasak1,21Faculty of Mechanical Engineering and Naval Architecture, Croatia,
2Wikki Ltd, United [email protected] [email protected],
Abstract
This work presents the Harmonic Balance methodfor incompressible non-linear periodic flows. Themethod is implemented and tested in foam-extend, a fork of the open source software Open-FOAM. Simulation results for passive scalar trans-port and NACA 2412 are presented and com-pared with a conventional transient simulation.Harmonic Balance for passive scalar transport isvalidated on four cases with simple harmonic andsquare waves. The Harmonic Balance Navier-Stokes solver is validated using NACA 2412 testcase in 2-D and 3-D.
1. Introduction
Harmonic Balance method is a quasi-steady statemethod developed for simulating non-linear tem-porally periodic flows. It is based on assumptionthat each primitive variable can be accurately pre-sented by a Fourier series in time, using first n har-monics and the mean value. Such assumptionsare used to replace the time derivative term withcoupled source terms, transforming the transientequations into a set of coupled steady state equa-tions. The improvement over steady-state meth-ods is that Harmonic Balance is able to describethe transient effects of a periodic flow without longtime-domain simulations.
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 momentum equation:
∇•(utjutj)−∇•(γ∇utj) = −2ω
2n + 1
2n∑i=1
P(i−j)uti
,•Harmonic Balance scalar transport equation:
∇•(uQtj)−∇•(γ∇Qtj) = −2ω
2n + 1
2n∑i=1
P(i−j)Qti
,•Harmonic Balance pressure equation:
∇•(
1
aP∇ptj
)= ∇•
(H(utj)aP
)=∑f
S.(H(utj)
aP
)f,
•Harmonic Balance continuity equation:
∇•utj = 0,
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 ddt term in its original form (pressureand continuity equation) remain the same, usingvariables corresponding to the time instant cur-rently calculated.
3. Numerical Procedure
Second order accurate, polyhedral Finite VolumeMethod is used. Segregated SIMPLE solution al-gorithm is adopted. Each of the 2n + 1 time stepsis resolved in its own SIMPLE loop.
4. Passive Scalar Transport
Passive scalar transport in 2D rectangular domainis simulated. Four test cases are presented, dif-fering in signal imposed on inlet:• Single sine wave• Two harmonic waves•Ramped square wave•Complex square wave
Inlet velocity is 10 m/s, diffusion coefficient is1.5 · 10−5 m2/s. This is valid for all the cases.Test cases are compared to transient simulationand data is extracted along the centreline of thedomain.
Single sine wave is resolved using one harmonic:
0 2.5 5 7.5 10Centerline length, m
-4
-2
0
2
4
Sca
lar
val
ue
Harmonic BalanceTransient
Figure 1: Left: scalar field visualisation at t = T .Right: scalar field comparison in t = T .
Two harmonic waves resolved using two har-monics:
0 2.5 5 7.5 10Centerline length, m
-6
-4
-2
0
2
4
6
Sca
lar
val
ue
Harmonic BalanceTransient
0 2.5 5 7.5 10Centerline length, m
-6
-4
-2
0
2
4
6
Sca
lar
val
ue
Harmonic BalanceTransient
Figure 2: Scalar field comparison in t = T/5,t = 2T/5.
Ramped square wave: solution converges usinghigher number of harmonics:
0 0.25 0.5 0.75 1
Time, s
-3
-2
-1
0
1
2
3
Scala
r valu
e
0 2.5 5 7.5 10Centerline length, m
-4
-2
0
2
4
Sca
lar
val
ue
TransientHarmonic Balance, 3 harmonics
Harmonic Balance, 5 harmonics
Harmonic Balance, 7 harmonics
Figure 3: Left: imposed signal. Right: solutionconvergence.
Complex square wave: solution converges usinghigher number of harmonics:
0 0.25 0.5 0.75 1
Time, s
-3
-2
-1
0
1
2
3
Scala
r valu
e
0 2.5 5 7.5 10Centerline length, m
-4
-2
0
2
4
Sca
lar
val
ue
TransientHarmonic Balance, 3 harmonics
Harmonic Balance, 5 harmonics
Harmonic Balance, 7 harmonics
Harmonic Balance, 10 harmonics
Figure 4: Left: imposed signal. Right: solutionconvergence.
5. NACA 2412 test case
Periodic airfoil pitching is simulated using 1, 3 and6 harmonics and compared to a transient simula-tion. Two cases are presented: 2D and 3D caseat Re = 1695. Comparison of pressure contoursaround the airfoil is given: 0–50 presents the lowercamber, 50–100 presents the upper camber.
2D, low Re case:
0 25 50 75 100Expanded airfoil cells
-0.2
0
0.2
0.4
0.6
Pre
ssure
, P
a
TransientHarmonic Balance
0 25 50 75 100Expanded airfoil cells
-0.2
0
0.2
0.4
0.6
Pre
ssure
, P
a
TransientHarmonic Balance
Figure 5: One harmonic comparison at t = T/3and t = 2T/3.
0 25 50 75 100Expanded airfoil cells
-0.2
0
0.2
0.4
0.6
Pre
ssure
, P
a
TransientHarmonic Balance
0 25 50 75 100Expanded airfoil cells
-0.2
0
0.2
0.4
0.6
Pre
ssure
, P
a
TransientHarmonic Balance
Figure 6: Three harmonics comparison at t =4T/7 and t = T .
0 25 50 75 100Expanded airfoil cells
-0.2
0
0.2
0.4
0.6
Pre
ssure
, P
a
TransientHarmonic Balance
0 25 50 75 100Expanded airfoil cells
-0.2
0
0.2
0.4
0.6
Pre
ssure
, P
a
TransientHarmonic Balance
Figure 7: Six harmonics comparison at t = 6T/13and t = T .
Figure 8: Flow field comparison at t = T .
3D, low Re case:
0 25 50 75 100Expanded airfoil cells
-0.4
-0.2
0
0.2
0.4
Pre
ssure
, P
a
TransientHarmonic Balance
0 25 50 75 100Expanded airfoil cells
-0.2
0
0.2
0.4
0.6
Pre
ssure
, P
a
TransientHarmonic Balance
Figure 9: One harmonic comparison at t = T/3and t = 2T/3.
6. Conclusion
The Harmonic Balance method is presented forpassive scalar transport and Navier-Stokes equa-tions. Scalar transport is validated using fourtypes of periodic impulses resembling sine wave,complex harmonic wave and two square waves.Harmonic Balance for Navier-Stokes equation isvalidated on NACA 2412 test case both for 2D and3D application.It is shown that the Harmonic Balance method is avaluable tool for tackling periodic problems in com-putational fluid dynamics.
10th OpenFOAM Workshop, June 29-July 2, 2015, University of Michigan, Ann Arbor
