acoustic modal analysis using cfd

Acoustic Modal Analysis using CFD M. Caraeni1, M. Oswald2

1 ANSYS Inc., Lebanon, NH 03766, USA, Email: [email protected]

2 ANSYS Germany GmbH, 64295 Darmstadt, Germany, Email: [email protected]

Abstract One of the most important issues in aeroacoustics design and development is the optimization of noise and vibration in both interior and exterior design processes. Especially the subjective comfort depends strongly on the interior noise. Besides the experimental investigations of the aeroacoustic phenomenon, numerical simulations have become more and more an effective tool to shorten the development cycle. This paper proposes a new approach based on a finite volume method for determining the resonance frequencies for enclosures. The current formulation requires as input the non-uniform mean flow obtained from a steady state solution together with prescribed boundary conditions. The method involves coupling the Linearized Navier-Stokes Equations with the Arnoldi algorithm. The Acoustic Modal Analysis, based on the iterative Implicitly Restarted Arnoldi method, offers a fast and efficient solver for large industrial eigenvalue problems. Using the data from this model, the natural acoustic modes of the system and the corresponding frequencies can be identified. These modes can resonate with interior acoustic source processes. The potential of the new method will be presented by means of several academic and industrial test cases ranging from thermo-acoustics to typical automotive and aerospace applications.

Introduction In many industrial applications there is the need to reduce

the acoustics related flow-field oscillations. Flows inside

enclosures are typical cases for resonance investigations.

Strong oscillatory phenomena can be found in the aerospace

industry, in the turbomachinery industry (combustion

chambers, etc.) and the automotive industry (car’s interior,

climate control ducts, etc.).

The Modal Analysis using CFD is a novel computational

tool, which is based on a Finite-Volume (FV) formulation. A

standard CFD simulation based on the compressible

Reynolds-Averaged Navier-Stokes equations (RANS)

provides the steady-state flow field with pressure, velocity

and temperature distributions. Depending on the steady state

results, the Modal Analysis computes the acoustic resonance

frequencies and the related acoustic (natural) modes in any

enclosure. Additionally, the model provides a stability

analysis of the computed modes. This paper presents the

theory of the Modal Analysis and a series of validation

cases.

Modal Analysis Theory The base equations of fluid mechanics (e.g. conservation of

mass, momentum and energy) are used to predict acoustic

modes and resonance frequencies. Only the mean flow

effects are taken into account to compute the acoustic

resonances.

First, the conservation of mass, momentum and energy are

expressed in differential form:

( )0=

∂∂+

∂∂

j

j

xu

tρρ

(1)

( ) ( )j

ij

ij

jii

xxp

xuu

tu

∂∂+

∂∂−=

∂∂+

∂∂ τρρ

(2)

( ) ( ) ( ) ( )Txu

xhu

te

j

iij

j

j ∇⋅∇+∂

∂=∂

∂+∂

∂ λτρρ (3)

where τij is the molecular stress tensor,

∂∂+

∂∂+

∂∂−=

i

i

i

j

k

kijij

x

u

x

u

x

u

2

1

32

δμτ (4)

Here, δij is the Kronecker delta function, μ is the molecular

viscosity, λ is the conductive heat diffusivity coefficient, cv

and cp are the specific heats at constant volume and constant

pressure, respectively,

21

1

2

22ii

vupu

Tce +−

=+=ργ

(5)

is the total energy per unit mass and

21

2iupp

eh +−

=+=ργ

γρ

(6)

is the total enthalpy per unit mass and γ is the ratio of the

specific heats.

Second, the linearized form of the conservation of mass,

momentum and energy can be achieved by superimposing

small perturbations on the mean flow. To model the

acoustics, this system of linearized Navier-Stokes equations

with appropriate boundary conditions is solved. These

equations can be written as:

00

0

0 =∂

′∂+∂∂′+

∂′∂+

∂∂′+

∂′∂

jj

j

j

j

j

j

xp

xu

xu

xu

tp ρρρ (7)

=∂∂′

−∂

′∂+∂∂′+

∂′∂+

∂′∂

iij

ij

j

ij

i

xp

xp

xu

uxu

ut

u 0

02

0

0

0

1

ρρ

ρ

∂∂+

∂∂+

∂∂′

−∂

′∂+∂

′∂+∂

′∂3

30

2

20

1

10

02

3

3

2

2

1

1

0

1

xxxxxx

iiiiii τττρρτττ

ρ (8)

( )10

0

0

0 −=∂∂′+

∂′∂+

∂∂′+

∂′∂+

∂′∂ γγγ

j

j

j

j

j

j

j

j

xu

pxu

pxp

uxp

utp

( ) ( ) ( )∂

∂′−∂

′∂−′∇⋅∇+∂′∂+

∂′∂

j

iji

j

iji

j

iji

j

iji

xu

xuT

x

u

x

u 0

0

00 ττλττ

(9)

NAG/DAGA 2009 - Rotterdam

737

The mean flow conditions are available

CFD simulation of the full Navier-Stoke

linearized form of the Navier-Stokes equ

compute the eigenvalues/eigenvectors of

coupled system of the Navier-Stokes eq

form can be written as:

0)(Re =+∂∂

Qst

Q

Where Q = (ρ, ρu, ρv, ρw, ρΕ) are the c

variables. With W = (p, u, v, w, T) as primi

can write:

0)(Re =+∂

∂∂∂

Wst

W

W

Q

Linearization presumes that the steady

averaged) solution W0 = (p0, u0, v0, w0, T0

with small perturbations of the primitive v

u´, v´, w´, T´). Fourier decomposition can

perturbation, when we consider a time p

solution. Equation (11) can be reformulated

( ) ( ) ( ) ( ) (ixWxtxWxtW +=+= ωexp,W, 0'

0

And after mathematical manipulations:

( ) ( ) ( ) ( ) 0'*Re'**

0 =∂

∂+∂∂

xWW

WsxW

W

Qiω

The Jacobians W

Q

∂∂

and ( )

W

Ws

∂∂ 0Re

linearizations of the discretized flow equati

With λ = -iω, this system of equations can b

( )[ ] [ ] 0=+− XAXBλ

[ ] XX λ=

where [ ] ( )W

WsA

∂∂

= 0Re, [ ] [ ] [ ]AB 1−= ,

( )xWX '= is the solution of the eigenvalu

the eigenvector for the complex eigenvalu

Analysis considers the coupled system o

including continuity, momentum, ener

transport. The proper acoustic response

included in the Modal Analysis. For furth

refer to [5].

The iterative Implicitly Restarted Arnoldi

based on ARPACK package is used to solv

problem (15). IRAM is a robust and efficie

numerical solution of a generalized eige

This model allows solving only for

eigenvalues and the corresponding eigenv

on the range of interest of an engineering

implementation in FLUENT® uses a s

which was formulated as a standard eig

The eigenvalue λ, λ= -iω, and the angul

both complex numbers. The sign of the

eigenvalue gives the stability of the corr

Reformulating W´:

from a precursor

es equations. The

uations is used to

the system. The

quations in vector

(10)

coupled conserved

itive variables, we

(11)

y-state (or time-

0) is superimposed

ariables W´ = (p´,

be applied on the

periodic perturbed

d into:

) ( )xWt ′⋅ω (12)

0 (13)

are the direct

ions.

be written as:

(14)

(15)

, [ ]W

QB

∂∂= and

ues problem. W´ is

ue λ = -iω. Modal

of equations (15),

rgy and species

of boundaries is

her details, please

i Method (IRAM)

ve the eigenvalues

ent method for the

envalues problem.

a small set of

vectors, depending

g application. The

hift-mode driver,

genvalue problem.

ar velocity ω are

e real part of the

responding mode.

( ) ( ) ( ) =′= xWtixt ωexp,W'

( ) ( ) ( )xWtti ri'expexp ⋅−⋅− λλ

one can show that the most un

the largest negative real part

real part indicates exponential

mode. The imaginary part

(frequency for mode’s oscillati

improves the convergence f

frequency spectrum. We can sh

scalar, λ0, and reformulate (15)

[ ] XX λ′=

with [ ] [ ] [ ] IAB 01 λ−= −

and

corresponding to eigenvalue λeigenvalues of this new s

transformed back to the eigen

Then, the frequencies of the

using:

( )πλλ

πλν

22

0iii +′==

Numerical results Box resonator

Simple resonator geometries ar

the fact that we can estimate

given temperature distribution

We consider two resonator ca

and the open resonator. The M

only the resonance frequencies

The domain of the closed box

length of 1m. All six side face

with constant temperature of 2

rest with the same temperature

the velocity perturbation is ma

at the two opposite walls. Henc

mode is equal to two times the

frequency for a box with afores

Hzc

f 1.1732

1.346 ===λ

The modal analysis calculates

(172.8 Hz). Figure 1 shows

frequencies of 173

Figure 1: Normalized pressurresonator (173 Hz and 345 Hz

(16)

nstable mode is represented by

eigenvalues (λr). A negative

ly increasing amplitude of the

indicates the angular speed

ion). The shift mode algorithm

for a desired range of the

hift the range of interest with a

) as:

(17)

d ( )xWX ′= , the eigenvector

0λλλ −=′ . After solving the

ystem, the eigenvalues are

nvalues of the original system.

original system are obtained

(18)

re good validation cases due to

the resonance frequencies for

ns and geometry dimensions.

ases: the closed box resonator

Modal Analysis calculates not

s, but also the acoustic modes.

resonator is a cube with a side

s are set to be wall boundaries

298 K. The box contains air at

e. For the first acoustic mode,

aximal in the middle and zero

ce, the wave length of the first

e box size. The first theoretical

said dimensions is therefore:

(19)

a frequency of about 173 Hz

the acoustic modes for the

Hz and 345 Hz.

re fluctuations for closed box )


738

A similar test case with one open side was

Modal Analysis. The open side was tre

outlet with static pressure 1 bar. The first

the mode for which the velocity perturbat

the open side. The first theoretical frequen

the foresaid dimensions is:

Hzc

f 5.864

1.346 ≅==λ

The frequencies calculated with the

algorithm match the theoretical value

frequency obtained with modal analysis: 8

shows the acoustic modes for the frequenc

354 Hz.

Figure 2: Normalized pressure fluctuationresonator (86 Hz and 354 Hz)

Double diaphragm

This is another case, where the coupling o

with the instantaneous flow field is imp

diaphragm is placed inside a circular duct w

0.05 m. The distance between the two diaph

1D. Turbulent eddies detached from the

impact the second diaphragm where stro

fluctuations are generated.

At the inlet the total pressure profile – co

Reynolds number of 15,000 - and the turbu

and ε, obtained from a previous LES run

The rotational periodic/symmetric geomet

good test case to investigate the effect

periodic boundary conditions on the moda

Analysis was performed on a full geome

cells and on a quadrant of the pipe with 7

symmetry as well as periodic boundary

predicted resonance frequencies and aco

identical (see table 1), hence Modal Anal

with periodic and symmetry conditions,

necessary simulation time. A comparison

based solver with the density-based s

identical results, which extends the ran

density-based solver is proper for all flo

incompressible to highly compressible

geometry case with 280,000 cells correspo

the resolution of the quadrant geometry w

count. The comparison between measured

and discrete acoustic modes is shown in F

the modal analysis will not give any i

s conducted using

ated as pressure-

acoustic mode is

tion is maximal at

ncy for a box with

(20)

Modal Analysis

very well (first

86.4 Hz). Figure 2

cies of 86 Hz and

ns for open box

of Modal Analysis

portant. A double

with a diameter of

hragms is equal to

e first diaphragm

ong wall pressure

orresponding to a

ulent profiles for k

[6], were applied.

try and flow is a

of symmetry and

al analysis. Modal

etry with 280,000

77,000 cells using

y conditions. The

oustic modes are

lysis can be used

which decreases

n of the pressure-

olver gives also

nge of use. The

ow regimes from

cases. The full

onds to four-times

with the same cell

frequency spectra

igure 5. Note that

nformation about

amplitude, hence no dB valu

modal analysis is presented at

3.

Figure 3: Comparison of m

experimental data at micro1 (be

(behind 2nd diaphragm)

Data 360°

PBNS

360°

DBN

336 Hz 353 Hz 353 H

553 Hz - -

805 Hz 747 Hz 747 H

1108 Hz 1055 Hz 1055

Table 1: Measured resonanc

computed frequencies for full g

(PBNS) and density-based solver

(periodic and symmetri

Figure 4 shows the unstable mo

1055 Hz.

Figure 4: Normalized pressdiaphragm

Combustion chamber G

Identification of acoustic eigen

obligatory prerequisite of co

[7]. GE LM 6000 is a swirl stab

combustor. A steady-state simu

incoming flow velocity of 150

and a static pressure of 517,10

The medium is a mixture con

H2, OH, O, O2, CO, CO2 and

ues or can be predicted. The

an arbitrary dB value in figure

ost unstable modes with the

etween diaphragms) and micro2

°

NS

90° per.

PBNS

90° sym.

PBNS

Hz 353 Hz 353 Hz

- -

Hz 747 Hz 747 Hz

5 Hz 1056 Hz 1055 Hz

e frequencies compared with

geometry (pressure-based solver

r (DBNS)) and quarter geometry

ic boundary condition)

ode at 353 Hz, 747 Hz and

sure fluctuations for double

GE LM 6000

nmodes of the combustor is an

ombustion instability research

bilized partially-premixed fuel

ulation was conducted with an

0 m/s at temperature of 617 K

07 Pa on a 250,000 cells mesh.

ntaining CH3, N2, HO2, H2O,

d CH4. Turbulence was solved


739

with the realizable k-ε turbulence model [

the modal analysis are summarized in table

Mode Frequ

First longitudinal 1L

Second longitudinal 2L

First tangential 1T

Second tangential 2T

First complex 1C

Second complex 2C

Table 2: Frequency and mode description

The lowest frequencies are acoustically d

sensitive to the effect of the combustion p

5 shows the first complex eigenmode at 346

Figure 5: Acoustic resonance mode at frequ

A full unsteady simulation with Large

(LES) was also performed. Fast Fourier Tr

been used for spectral analysis. To

frequencies, the LES has to run for long flo

required resolution of the frequency spe

frequency range the modal analysis has a

transient methods. Figure 6 shows the insta

magnitude in the symmetry plane of

chamber. Frequencies less than 7000 Hz a

the modal analysis model in figure 7.

Figure 6: Instantaneous vorticity magnitudthe symmetry plane of combustion chambesimulation

Figure 7: Discrete frequency spectrum resuusing the CAA approach compared with MA

[8]. The results of

2.

uency [Hz]

80

180

980

1973

3462

4675

n of combustor

driven and are not

process [9]. Figure

62 Hz.

uency 3427 Hz

e-Eddy-Simulation

ransformation has

predict the low

ow time to get the

ectra. In the low

an advantage over

antaneous vorticity

the combustion

are compared with

e distribution in r from unsteady

ults as computed A results

Summary A fast finite-volume based m

computing the resonance frequ

any complex enclosures is pres

This fast and accurate numeri

non-uniform flow field obtaine

computation. Based on th

equations with appropriate

generalized eigenvalue probl

solved using IRAM.

The Modal Analysis method b

orders of magnitude more

approach, of computing the

direct unsteady simulation.

References [1] Wind-tunnel experiments

cavities at subsonic and tr

Technical Report 3438, A

Reports and Memoranda,

[2] On the tomes and pressur

flow over rectangular cav

P.J.W., Fluid Mechanics, V

[3] Modelling and predictio

amplitude and frequency

McCotter F., Technical Re

[4] Estimation of possible exc

shallow rectangular cavitie

AIAA Journal, Vol. 11, pp

[5] Efficient acoustic modal an

Caraeni M. et al, AIAA-20

[6] Simulation of Aeroacousti

Control Systems, Mathey F

AIAA-2006-2493,2006

[7] Prediction and Control of C

Industrial Gas Turbines, K

Thermal Engineering 24, 1

[8] A New k-ε Eddy-Viscosity

Number Turbulent Flows -

Validation. Shih T.H. et al

24(3):227-238, 1995.

[9] Low-frequency combustio

vortices. Pointsot T. et al,

International de l’Energy,

method, based on IRAM, for

uencies and acoustic modes of

sented.

ical method uses as input the

ed from a previous steady-state

he linearized Navier-Stokes

e boundary conditions a

lem is formulated, which is

based on IRAM proves to be

efficient than the classical

resonance frequencies from

on the flow over rectangular

ransonic speeds. Rossiter J.E.,

Aeronautical Research Council

1964

re oscillations induced by the

vities. Tam C.K.W. and Block

Vol.89 pp. 373-399, 1978

n of weapons bay acoustic

y. Cain A.B., W.W. Bower,

eport, VEDA Inc., 1996

citation frequencies for

es. Covert E.E., Bilanin A.J.,

p 347-351, 1973

nalysis for industrial CFD.

009-1332, 2009

ic Sources in Aircraft Climate

F, Morin O., Caruelle B.,

Combustion Instabilities in

Kelsall G., Troger C., Applied

1571-1582, 2004.

y Model for High Reynolds

- Model Development and

l, Computers Fluids,

on instabilities driven by large

Conference de l’Agence

Heidelberg, 1986


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acoustic modal analysis using cfd

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