Fundamentals of Fan Acoustics

8/9/2019 Fundamentals of Fan Acoustics

1/97

Fundamentals of Fan Aeroacoustics


2/97

Overview of Lecture

• Noise Sources and Generation Mechanisms – Sources of Noise for Typical Fans – Fluid-Structure Interaction as a Noise Generation Mechanism – Coupling to the Duct: Propagating Modes and Cut-off Phenomena

• Modeling of Fan NoiseThe Acoustic AnalogyComputational Methods: Aeroacoustics and UnsteadyAerodynamicsThe Linear Cascade ModelEffects of Geometry and Blade Loading on Acoustic Radiation

• Recent Developments in Fan Modeling – Tonal and Broadband Noise – Nonuniform Mean Flow Effects:swirl – Three-Dimensional Effects – High Frequency Effects

• Conclusions


3/97

Dominant Noise Sources for Turbofan Engines


4/97

High Bypass Ratio Fan Flyover Noise

Maximum Perceived Noise Level

Approach Takeoff


5/97

Typical Turbomachinery Sound Power Spectra

Subsonic Tip Speed Supersonic Tip Speed


6/97

Turbomachinery Noise Generation Process


7/97

Rotor-Stator Interaction


8/97

Fluid-Structure Interaction as aNoise Generation Mechanism

• The interaction of nonuniform flows with structural components such as

blades and guide vanes produce fluctuating aerodynamic forces on the bladesand radiates sound in the farfield.

• Noise Sources: Flow Nonuniformities: Inlet Turbulence, Boundary Layers,

Tip and Hub Vortices,Wakes etc .

• Mechanism: Interaction with Rotating Components (rotor noise), Scattering bySharp Edges (trailing edge noise), Impingement of Unsteady Nonuniformitieson Guide Vanes ( rotor/stator interaction).

• Propagation in the Duct: Sound Must Propagate in a Duct: therefore only highfrequency acoustic modes will propagate.


9/97

Rotor Wake Phenomena


10/97

Rotor Wakes Interaction with Downstream Stator


11/97

Rotor-Stator Interaction

Wakes and Tip Vortices


12/97

Scaling Analysis

• Multiple length scales: – Duct hub and tip Radii: R h, R t, Rotor/stator spacing: L – Chord length c, Blade spacing s=2pR/(B or V) – Turbulence Integral Scale = Λ


13/97

Multiple Pure Tones


14/97

Airfoil in Nonuniform Flow


15/97

Equations of Classical Acoustics

0')1(

0'1

0'

'

'

2

2

2

0

2

0

0

0

0

0

=∂∂−∇

=∇+∂

∂=⋅∇+∂∂


16/97

Acoustic Intensity and Energy

• Fundamental Conservation Equation

u p I

u p E

I t

E

rr

r

'22

''

0

20

0

=

+=

=⋅∇+∂

∂

ρ

ρ

ρ


17/97

Fundamental Solution of the wave Equation

• Spherical symmetry:

• Green’s Function

t-R/c: retarded time• Compact/Noncompact

Sources

y-xR ,)( rrm ==Φ

Rc

Rt f

R

)cR t(

),y|t,x(G−τ−δ

=τrr

Causality determines the sign: Soundmust propagate away from the source


18/97

Plane Waves

00

2

2

1

00

2''

00

2'2

20

02''

0

')(

00

)('

c

p I

k k

c pu p I

c

pu p E

k

k

c

pt xk ie

k

k

c

pu

t xk ie p p

ρ

ρ

ρ ρ

ρ ρ

ρ

ω

ρ

ω

=

==

=+=

=−⋅=

−⋅=

r

rr

rrr

rr

rr


19/97

Elementary Solutions of the Wave Equation

No Flow

Noise Source Acoustic Pressure Acoustic Intensity Directivity

Point Source Sperical symmetry

Dipole: Force

Quadrupole: Stress

c Rc

Rt m

20

2

2..

16

)(

ρ π

−

Rc Rt m

π 4)/(

..−

Rc4

e).c/R t(F r .

π

− rr θ cos

dt

dF

dt

dF

Rc

x x ji ji

432

16 ρ π

234

..

c R

j xiVxijT

π 462

2....

16 c R x x x xV T T lk jiklij

ρ π θ θ sincos


20/97

Directivity of Elementary Sources of sound

Monopole Directivity Dipole Directivity Quadrupole Directivity


21/97

Scaling of of Acoustic Power Radiated for

Aerodynamic Applications at Low Mach Number

Source Scaling Ratio

Dipole 1

Quadrupole 2

6

8

Thus at low Mach number dipoles or forces are more efficient sources of noise thanquadrupoles or turbulence. However, this result is valid only for uniform flows and lowfrequency.


22/97


23/97

Acoustic Waves in Ducts

Square Duct• Higher order modes

• Dispersion Relation

• Propagating or Cut-on Modes – k mn:realEvanescent Modes

– k mn: imaginary

[ ])()(3cos2cos' 11 t mnk iemn Bt mnk iemn Aa xm

a

xm p x x ω ω π π +−+−=

)nm(ac

k 2222

2

2

mn +π−ω=

Plane waves always propagate.Higher order modes propagate onlywhen

2aora

c ω

The velocity and pressure ofevanescent or cut-off modes areout of phase and there is no net

transport of energy (I=0).


24/97

Phase and Group Velocities for

Dispersive Waves

• Phase Velocity

– Phase velocity is largerthan the speed ofsound

• Group Velocity

– Group velocity is thevelocity at which

acoustic energy istransported in the duct.It is smaller than thespeed of sound.

22

2222 )(1

ω π

ω

ac

nm

ck

c ph+−

==22

2222 )(1

ω π

ω a

cnmc

dk d

cg +−==


25/97

Higher Acoustic Modes in a Duct

• Reflecting wave components making up the first higher mode propagating between two parallel plates. Solid lines represent pressure maxima of the wave; dotted lines, pressure minima.Arrows, representing direction of propagation of the components,

are normal to the wavefronts.


26/97

Application to Rotor/Stator InteractionThe Tylor-Sofrin Modes

• Incident Gust

– Interblade phase angle

• Cut-on Modes

m=m′-qV

• For rotor/stator interaction with B rotor blades and V guide vanesm ′ =pB,

• Hence – The circumferential modal number m for propagating modes for a rotor with B blades and a

stator with V blades is given by

m=pB-kV

Example: B=18, V=40. m=-22, -4, 10, 14, 22

U

k ,e)r (au g)t'mxk (i)g(

mgg ω== ω−θ+

rr

V mm

π ϑ σ

2'' ==

∑ ω−θ+=

n,m

)tmxk (in,ma

mne)r (au rr


27/97

• The governing equation

• With the boundary condition at the hub and the tip

• The eigenvalues are given by,

Where

• The solution is given by

Sound Propagation in a Duct with Uniform Flow

0=∂

′∂

r

p

02220 =′∇+′ p

t D

p D

( )t m xk imn

m nn

mner K p ω θ γ −+∞

−∞=

∞

=∑ ∑=′ )(

0

)1())(1(~~

2

2222

x

mn x xmn M

m M M k

−+−−±−= γ ω ω

smM −=ω ω ~


28/97

Conclusions

• Fan noise sources: Flow nonuniformities andirregular flow pattern.

• Mechanism: Fluid/structure interaction and link tounsteady aerodynamics.

• Classical acoustics concepts are essential tounderstanding and modeling of noise.

• The coupling to the duct determines the modalcontent of the scattered sound and affects sound propagation.


29/97


30/97

Modeling of Fan Noise

• The Acoustic Analogy• Computational Methods: Aeroacoustics

and Unsteady Aerodynamics

• The Linear Cascade Model• Effects of Geometry and Blade Loading

on Acoustic Radiation


31/97

Lighthill’s Acoustic Analogy

• Inhomogeneous wave equation

• Lighthill’s stress tensor

• For a Uniform Mean Flow

ji

ij

x xT c

t ∂∂∂=∇−

∂∂

222

02

2 ρ ρ

ijij jiij ec pvvT −−+= )(20 ρ δ ρ

k

k ij

i

j

j

iij x

v x

v

xv

e∂∂−

∂∂

+∂∂= μδ μ

32)(

ji

ij

x x

T c

t D

∂∂∂

=∇−∂

222

02

20 ρ

ρ


32/97

Aeroacoustics and Unsteady Aerodynamics

• Sound is the far-field signature of the unsteady flow.

• Unsteady aerodynamics has been developed for aeroelastic problemssuch as flutter and forced vibrations where the main interest is todetermine the near-field body surface forces.

• The aeroacoustic problem is similar to that of forced vibration butwith emphasis on the far-field. It is a much more difficultcomputational problem whose outcome depends on preserving the far-field wave form with minimum dispersion and dissipation.

• Inflow/outflow nonreflecting boundary conditions must be derived tocomplete the mathematical formulation as a substitute for physicalcausality.


33/97

Airfoil in Nonuniform Flow


34/97

Disturbaces in Uniform Flows

Splitting Theorem:

The flow disturbances can be split into distinct potential(acoustic),vortical and entropic modes obeying three independent equation.

– The vortical velocity is solenoidal, purely convected andcompletely decoupled from the pressure fluctuations.

– The potential (acoustic) velocity is directly related to the pressurefluctuations.

– The entropy is purely convected and only affects the densitythrough the equation of state.

– Coupling between the vortical and potential velocity occursonly along the body surface.

– Upstream conditions can be specified independently for variousdisturbances.

S litti g f th V l it i t A ti


35/97

Splitting of the Velocity into Acoustic,

Entropic and Vortical Modes

Dt

D' p

.U tDt

D

0'sDtD

0)Dt

D

c

1( 0' puDt

D

0uDtD 0u'

DtD

uu )t,x(uU)t,x(V

00

0

0

2

2

20

200

0

v0

00

v

φρ−=

∇+∂∂≡

=

=φ∇−=⋅∇ρ+

==⋅∇ρ+ρ

φ∇+=+=

r

r

rr

rrrrrrr


36/97

Equations for Linear Aerodynamics

• Vortical Mode:• Harmonic Component

• Potential Mode:

• Boundary Conditions: impermeability along blade surface, Kuttacondition at trailing edge, allow for wake shedding in response to gust.

)( iU xuvurrrr −∞=

)( t xk ieag

u ω −⋅=rr

rr

0)22

2020

1( =∇− φ Dt

D

c


37/97

Flat Plate in a Gust

)( t xk ieagu ω −⋅=rr

rr

Transverse Gust: (0,a2,0), (k 1,0,0)

Oblique Gust:(0,a2,0), (k1,0,k3)

V Di Sh i h R l d I i P f h R


38/97

Vector Diagram Showing the Real and Imaginary Parts of the Response

Function S(k 1,0,M) versus k 1 for a Transverse Gust at Various M


39/97

Vector Diagram Showing the Real and Imaginary Parts of the ResponseFunction S(k 1,k3,0.8) versus k 1 for a Transverse Gust at Various M


40/97

Airfoil in Three-Dimensional Gust


41/97

Vector Diagram Showing the Real and Imaginary Parts


42/97

Vector Diagram Showing the Real and Imaginary Partsof the Response Function S(k 1,k2,k3) versus k 1 for an

Airfoil in an Oblique Gust (k 2=k1) at Various k 3.

Acoustic Directivity for a 3% Thick Airfoil in aTransverse Gust at M=0 1 k =1 0


43/97

Transverse Gust at M=0.1, k 1=1.0-.-, direct calculation from Scott-Atassi’s code; , Kirchhoff’s

method;---, flat plate semi-analytical results.

Acoustic Pressure Directivity for a Symmetric Airfoil in a


44/97

Two-Dimensional GustThichness Ratio=0.06, M=0.7, k 1=3.0

k2: solid line,0.0; ------,3.0.

Total Acoustic PressureDipole Acoustic Pressure

Acoustic Pressure Directivity for a Lifting Airfoilin a transverse Gust


45/97

Total Acoustic PressureDipole Acoustic Pressure

in a transverse GustThichness Ratio=0.12, M=0.5, k 1=2.5

Camber: solid line,0.0; ------,0.2; -- . --, 0.4.


46/97


47/97

The Linear Cascade Model

• Separate rotor and stator and consider each blade rowseparately.

• Unroll the annular cacade into a linear cascade of infinite bladeto preserve periodicity.

• Flat-plate cascade : uniform mean flow, integral equationformulation in terms of plane waves. The current benchmark.

• Loaded cascade : Linearized Euler about a computationallycalculated mean flow, requires field solutions of pde.CASGUST and LINFLOW are current benchmarks.


48/97


49/97

Linear Cascade and Strip Theory


50/97

Cascade of Airfoils in a Three-Dimensional Gust


51/97


52/97


53/97


54/97

Magnitude of the Response Function S versus k1 for anEGV C d ( ) ith Fl t Pl t C d ( i l )


55/97

EGV Cascade( squares ) with Flat-Plate Cascade( circles ).

M=0.3, k2=k1


56/97


57/97

How Good is the Linearized Euler Model?


58/97

Cascade Flow with local Regions of Strong Interaction


59/97


60/97

Magnitude (a) and Phase (b) of the first Harmonic Unsteady PressureDifference Distribution for the subsonic NACA 0006 Cascade Underhoing

an In Phase Torsional Oscillation of Amplitude a=2o at k1=0 5 about


61/97

an In-Phase Torsional Oscillation of Amplitude a=2o at k1=0.5 about

Midchord; M=0.7; ,Linearized Analysis; …. Nonlinear Analysis.

Magnitude (a) and Phase (b) of the first Harmonic Unsteady PressureDifference Distribution for the subsonic NACA 0006 Cascade Underhoing

I Ph T i l O ill i f A li d 2 k1 0 5 b


62/97

an In-Phase Torsional Oscillation of Amplitude a=2o at k1=0.5 about

Midchord; M=0.7.

Conclusions


63/97

Conclusions

• Cascade effects such as blade interference which depends on thespacing ratio, and stagger have strong influence on the aerodynamicand acoustic cascade response, particularly at low frequency.

• At high frequency the cascade response is dominated by the acoustic

modes cut-on phenomena.• For thin blades, the leading edge dominates the aerodynamic pressureand noise generation.

• For loaded blades at high Mach number, large unsteady pressureexcitations spread along the blade surface with concentration near thezone of transonic velocity.

• Blade loading changes the upstream and downstream flows and thusaffects the number and intensity of the scattered sound.

Recent Developments in Fan Modeling


64/97

Recent Developments in Fan Modeling

• Tonal and Broadband Noise• Nonuniform Mean Flow Effects : swirl• Three-Dimensional Effects• High Frequency Effects

Tonal and Broadband


65/97

Tonal and Broadband

• Turbulence modeling using the rapiddistortion theory.

• Hanson (Pratt & Whitney), Glegg (Florida)developed models using linear flat-platecascades.

• Effects of blade loading, 3D effects areunder development at ND

Turbulence Representation


66/97

p

• Fourier representation:

• Evolution of each Fourier component

• Velocity covariance

• One-dimensional energy spectrum

)'()()'()(

),(),( ).(,

k k k k ak a

d k d ek at xu

ij ji

t xk i

k

rrrrr

rrrrr rr

r

−Φ=

= −∫δ

ω ω ω

ω

Utik jiji

1ea)k ,x()t,x(u −μ= rrr

k d e)k ()k ,'x()k ,x()t,'x,x(R Utik )0(k jk

*ik ij

1rrrrrrrr

llr

−Φμμ= ∫

32)0(

k j*ik 1ij dk dk )k ()k ,'x()k ,x()k ,'x,x(

rrrrrrrll Φμμ=Θ ∫ ∫

+∞

∞−

+∞

∞−

Aerodynamic and Acoustic Blade Response


67/97

Swirling Meanflow +

disturbance

Normal mode analysis“ construction of nonreflecting

boundary conditions”

Linearized Euler model

Rapid distortion theory“ disturbance propagation”

Blade unsteadyloading & radiated

sound field

Source termon blades

Non-reflectingboundary

conditions

Mathematical Formulation


68/97

• Linearized Euler equations

• Axisymmetric swirling mean flow

• Mean flow is obtained from data or computation• For analysis the swirl velocity is taken

• The stagnation enthalpy, entropy, velocity and vorticityare related by Crocco’s equation

θ+= e)r ,x(Ue)r ,x(U)x(U sxxrrrr

r r U s

Γ+Ω=

ζU ×+∇=∇ STH

Normal Mode Analysis


69/97

y

• A normal mode analysis of linearized Euler equations

is carried out assuming solutions of the form

•• A combination of spectral and shooting methods is

used in solving this problem – Spectral method produces spurious acoustic modes

– Shooting method is used to eliminate the spuriousmodes and to increase the accuracy of the acousticmodes

)xk mt(i mne)r (f +θ+ω− ν

r mU

Uk DtD s

xmn0

mn ++ω−==Λ

Mode SpectrumSpectral and Shooting Methods


70/97

Mx=0.55, MΓ=0.24, MΩ=0.21, ω=16, and m=-1

Pressure Content of Acousticand Vortical Modes


71/97

Mx=0.5, MΓ=0.2, MΩ=0.2, ω=2π, and m=-1

Effect of Swirl on Eigenmode Distribution


72/97

Mxm=0.56, MΓ=0.25, MΩ=0.21

Summary of Normal Mode Analysis


73/97

Pressure-Dominated Acoustic Modes

Vorticity-Dominated Nearly-Convected Modes

Propagating DecayingSingular Behavior

Normal Modes

NonreflectingBoundary Conditions

Nonreflecting Boundary Conditions


74/97

I n f l o w

C o n d

i t i o n s

O u

t f l o w

C o n

d i t i o n s

C o m p u

t a t i o n a l

D o m a i n

Quieting the skies: engine noise reduction for subsonic aircraftAdvanced subsonic technology program. NASA Lewis research center, Cleveland, Ohio

• Accurate nonreflecting boundary conditions are necessary forcomputational aeroacoustics

Formulation


75/97

• Only outgoing modes areused in the expansion.

• Group velocity is used todetermine outgoing modes.

ω ω θ ω

ν ω

ν d er pct x p xk mt i

nmnmn

mn )(

0),(),( ++−

∞

=

∞

−∞=∑∑∫=

r

Causality

• Presssure at the boundaries is expanded in terms of theacoustic eigenmodes.


76/97

Rotor/Stator Interaction


77/97

The rotor/stator system isdecoupled

The upstream disturbance can bewritten in the form,

Quasi-periodic conditions

Wake discontinuity

Nonreflecting conditions

)tm(i

mm

)tm(i

mmI

e)r (A),r ( p

e)r (a),r (u

I

ω−θ′∞

−∞=′′

ω−θ′∞

−∞=′′

∑

∑=θ

=θ rr

Numerical Formulation


78/97

The rotor/stator system isdecoupled

The upstream disturbance can bewritten in the form,

Quasi-periodic conditions

Wake discontinuity

Nonreflecting conditions

)tm(i

mm

)tm(i

mmI

e)r (A),r ( p

e)r (a),r (u

I

ω−θ′∞

−∞=′′

ω−θ′∞

−∞=′′

∑

∑=θ

=θ rr

Tip

Hub

Two Vanes

Computational Domain

Domain DecompositionInner and Outer Regions


79/97

Inner and Outer Regions

Vorticity and PressureAre uncoupled.

Vorticity and Pressure are Coupled.Solution is Simplified by Removing Phase.

O(R) O(c)


80/97

Scattering Results

Parameters for Swirling Flow Test Problem


81/97

Narrow Annulus Full Annulus Data

r tip/r hub 1.0/0.98 r tip/r hub 1.0/0.75 M(mach number)

0.5

α (disturbance) 0.1B (rotor blades) 16

C (chord) 2π/VStagger 45o

V (stator blades) 24

L (length) 3c

ω 0.5π,1.0π1.5π,2.0π2.5π,3.0π

3.5π,4.0π

ω 3.0π

Computational Domain


82/97

B (rotor blades) 16

V (stator blades) 24C (chord) 2π/

VL (length) 3c

( )nBc

r nB

o

m O≈Ω=ω

Narrow Annulus Limit


83/97

r tip/r hub 1.0/0.98ω 1.5π, 4.5π, 6.5πM 0.5Stagger 45

Narrow Annulus Limit


84/97

Gust Response for Narrow Annulus Limit –Comparisonwith 2D cascade

Mo=0.3536, Mθ=0.3536, B=16, V=24, ar =0


85/97

Unsteady Lift Coefficient Acoustic Coefficientsm=-8, n=0,1


86/97

Mean Flow Cases


87/97

Mean Flow 1: Mo=0.4062, MΓ=0., MΩ=0.

Mean Flow 2: Mo=0.3536, MΓ=0.2, MΩ=0.Mean Flow 3: Mo=0.3536, MΓ=0., MΩ=0.2Mean Flow 4: Mo=0.3536, MΓ=0.1, MΩ=0.1

Effect of Mean Flowr h\r t=0.6, ω=3π, and ar =0


88/97

Unsteady Lift Coefficient Acoustic Coefficients

Mode m=-8 MeanFlow 1MeanFlow 2

MeanFlow 3

Meanflow 4

Downstreamn=0

0.2015 0.1610 0.1375 0.1349

Downstreamn=1 Cut off 0.0787 0.2140 0.1698

Upstreamn=0

0.1363 0.0143 0.0313 0.0195

Upstream

n=1Cut off 0.0370 0.0763 0.0586

Effect of FrequencyMean Flow 4


89/97

Unsteady Lift Coefficient

Mo=0.3536, MΓ=0.1, MΩ=0.1, r h / r t =0.6667, and ar =0

Acoustic Coefficients

m=-8, n=0,1


90/97

Scattering of Acoustic versus VorticalDisturbance


91/97

Unsteady Lift Coefficient Acoustic Coefficients

Mo=0.3536, MΓ=0.1, MΩ=0.1, r h / r t =0.6667, and ω=3π

Mode m=-8Vortical

DisturbanceAcoustic

Disturbance

Downstreamn=0

0.1471 0.8608

Downstreamn=1

0.0908 0.1458

Upstreamn=0

0.0185 0.0718

Upstreamn=1 0.0498 0.0775

Comparison of Successive Iterations


92/97

Three-Dimensional EffectsComparison With Strip Theory


93/97

Effect of Swirl


94/97

Effect of Blade Twist


95/97

Conclusions


96/97

• Swirl affects the impedance of the duct and the number of

cut-on acoustic modes. The spinning modes are notsymmetric.• Strip theory gives good approximation as long as there is

no acoustic propagation.• Resonant conditions in strip theory are much more

pronounced than for 3D, i.e., lift variation in 3D is muchsmoother along span.

• Discrepancies between strip theory and 3D calculationsincrease with the reduced frequency: it is a high frequency phenomenon.


97/97

Documents

Fundamentals of Fan Acoustics