Harmonic Balance Method forUnsteady Periodic Flows

Gregor Cvijetić1, Hrvoje Jasak1,21Faculty of Mechanical Engineering and Naval Architecture, Croatia,

2Wikki Ltd, United Kingdom1gregor.cvijetic@gmail.com 2h.jasak@wikki.co.uk,

hrvoje.jasak@fsb.hr

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