Elementary non-linear processes

in aero-acoustics of internal flows

Avraham (Mico) Hirschberg

Technische Universiteit Eindhoven

Workshop N3L, 17-20 may 2010

TU München

Outline

1. Wave distortion 1-D frictionless

(shock waves)

2. Driven resonance in closed pipes (sub-harmonic resonance, turbulence, thermal effects)

3. Driven resonance in open pipes

(vortex shedding)

4. Self-sustained oscillation (Rijke tube)

fpvvt

v rrrrrr

+⋅∇+−∇=

∇⋅+

∂

∂σρ )(

0)( =⋅∇+∂

∂v

t

rρ

ρ

s

s

pc

ss

pppspp

∂

∂=

∂

∂+

∂

∂=⇒=

ρ

ρρ

ρρ

2

'''),(

Mass conservation

Newton

Local thermodynamic equilibrium

x

p

x

uu

t

u

∂

∂−=

∂

∂+

∂

∂ρ

0=∂

∂+

∂

∂+

∂

∂

x

u

xu

tρ

ρρ

∂

∂+

∂

∂=

∂

∂+

∂

∂

xu

tc

x

pu

t

p ρρ2

Frictionless, Isentropic and One-dimensional

( )),();,(

);,(;0,0),,(

txcctx

txpptxuv

==

==

ρρ

r

1-D:

Equations of motion

Isentropic flow

x

p

c

c

x

uu

t

u

∂

∂−=

∂

∂+

∂

∂

ρ

0][2

=∂

∂+

∂

∂+

∂

∂

x

uc

xu

tc

c ρρ

ρ

∂

∂+

∂

∂=

∂

∂+

∂

∂

xu

tc

x

pu

t

p ρρ2

Characteristic form: step 1

Eliminate density

x

p

c

c

x

uu

t

u

∂

∂−=

∂

∂+

∂

∂

ρ

0][1

=∂

∂+

∂

∂+

∂

∂

x

uc

x

pu

t

p

cρ

Characteristic form

0)( =

±

∂

∂±+

∂

∂∫ c

dpu

xcu

t ρ

Adding or subtracting

Sound waves

U

xU + cU - c

C+

C-

Characteristics

cudt

dxC −=−

:

+C

−C

Along the lines in the (x,t) diagram

We follow either a C+ line or a C- line.

x

t

∫−=Θ−

c

dpu

ρ

cudt

dxC +=+

:

∫+=Θ+

c

dpu

ρ

Initial value problem

),()0,(

),()0,(

2

1

txx

txx

c

dpu

c

dpu

c

dpu

c

dpu

−=

−=Θ

+=

+=Θ

∫∫

∫∫

−

+

ρρ

ρρ

+C−C

x

t

1 2

We find u(x,t) and at the intersection of the

two characteristics C+ and C- .∫ cdp ρ/

(x,t)

Simple wave:

travelling into a uniform region

+Θ=

=+⇒

−Θ=

−=−

Θ=+

∫∫

∫

∫∫

∫

+

+

−

++

0

0

0

2

1

2

1

:

:

c

dp

c

dp

constcu

c

dpu

c

dp

c

dpuC

c

dpuC

ρρ

ρ

ρρ

ρ tcx 0=

+C

x

t

Uniform region

straight line +C

Fundamental derivative

s

s

p

c

dccdc

c

p

cc

dp

∂

∂+=Γ

−Γ=

∂

∂=

2

2

21

12

1

ρ

ρρ

Simple wave in calorically perfect gas:

Boundary condition p(0,t)

−

−=

=

−

1),0(

1

2),0(

;),0(

),0(

0

0

2

1

0

0

c

tcctu

p

tpctc

γ

γ

γ tcx 0=+

C

x

t

Uniform region

Both u and c increase with increasing pressure because 1>γ

2

1+=Γ

γ

Wave propagation non-linearity

ts

Single valued solution fails for t>ts

Compression wave:

Boundary condition p(0,t)

( )dtcdtudtt

x

ct

x

s

s

s

s

,0),0(

0

+=−

=

For t > ts we have a multiple valued solution!

),( ss tx

x

t

dt

C+2

C+1

Simple wave:

Boundary condition p(0,t)

( )

++−=

+=−

=

dtdt

dp

dp

cudc

t

dt

t

x

dtcdtudtt

x

ct

x

ss

s

s

s

s

s

)()1(

,0),0(

0

0

),( ss tx

x

t

dt

Simple wave:

Boundary condition p(0,t)

00

200

/1

2

)/(

+≈

Γ=

dtdp

p

dtdp

cts

γ

γρ

),( ss tx

x

t

dt

Brassy sound (Beauchamp, Hirschberg, Msallam)

mouth piece pressurepipe pressure

Lips of Murray CampbellPressure in mouth piece

Pressure in trombone

Shock wave

Integral conservation laws across a

shock moving with the shock.

2

)(

2

)(

)()(

)()(

2

22

2

11

2

2

221

2

11

2211

ss

ss

mss

uuh

uuh

puupuu

uuuu

−+=

−+

+−=+−

=−=−

ρρ

φρρ

12

shock

Rankine Hugoniot (RH)

• Eliminate the velocity

12

12

12

2

1

2

2

12

12

122

)11

)((

112

11

hhpp

hhppm

−=+−⇒

−

−−=

−

−−=

ρρ

ρρρρ

φ 12

Comparison of RH with isentrope

−

=

−==

−=+−

v

p

c

ss

p

p

pTch

hhpp

0

00

12

12

12

exp

1

)11

)((

γ

ρ

ρ

ργ

γ

ρρ

ρ

1

p

max

1

ρ

Weak shock

...)()/1(

12

1 3

122

2

1

12 +−

∂

∂=

−pp

pTcc

ss

svv

ρ

ρ

1

p

max

1

ρ

Weak shocks are almost adiabatic

Speed of weak shock wave

x

t

)]()[(2

12211 cucuus +++≈

1

2

Weak shock: change in (u+c)

+

1C

+

2C

−C

x

t

ρρ

ρρ

ρρ

ρρ

dc

cud

dp

cc

c

dp

p

c

c

dpcud

c

dpdu

c

dpudC

ss

Γ=+

∂

∂+=

∂

∂+=+

=−=− ∫−

)(

21

2)(

0)(:

22

Weak shock speed

+

1C

+

2C

−C

x

t

( ) ...2

12

1

0

...

)(

11122

1

1122

+

∆Γ+=+++=

=

+∆Γ

++=+

Γ=+

ρ

ρ

ρρ

ρρ

ccucuu

u

ccucu

dc

cud

s

Shock waves in aircraft noise

K.L. Gee et al (JASA 2008)

Weak shock theory (Beyer 1997, Pierce 1981…)

• Neglecting the effect of entropy change on the wave propagation

• Predicting shock attenuation due to friction and heat transfer in the shock wave

• Sawtooth solution

• N-waves

• Aircraft noise (Perception)

Limit of wave shape (Sawtooth)

00 ρρ ∆=∆ LLx

t

L

ρ∆

Mass (we neglect entropy changes across the shock)

Weak shock:

area rule (Landau 1942-1945)

L

L

LLL

00

0012 )(

ρρ

ρρρρ

∆=∆

∆=∆=−

ρ

1c

22 cu +

xL

)(2

1122 ccuus ++=

Mass conservation

Weak shock: attenuation (Landau 1942-1945)

L

Lcccu

dt

dLs

00111

22)(

ρ

ρρ

ρ ∆Γ=

∆Γ=−=

ρ

1c

22 cu +

xL

)(2

1122 ccuus ++=

Weak shock: attenuation (Landau 1942-1945)

tL

L

L

tcLL

1

21

00

0

100

∝∆=∆

Γ∆+=

ρρ

ρ

ρ

ρ

1c

22 cu +

xL

)(2

1122 ccuus ++=

Summary weak shock wave

Equal area rule

x

t

)]()[(2

12211 cucuus +++≈

1

2

Shock structure and viscous

damping (Lighthill 1956)

2

2

)(x

u

c

dpu

xcu

t ∂

∂=

±

∂

∂±+

∂

∂∫ δ

ρ

Burgers equation!

Viscous damping (Chester 1964))( 2 suu − )( 1 suu −

2

2

02 )(y

u

x

uc

x

uuu s

∂

∂=

∂

∂≈

∂

∂− ν

δv

Linear theory for growth viscous boundary layer,

allows estimate of viscous force.

ss uxuuuxu −=−=∞ )0,(;),( 2

Viscous force

)(21

2

)(|

00

2

00

2

xtcHx

cu

Rx

p

x

uu

t

u

RSf

xtcHx

cu

y

u

w

d

wx

ww

−−≈∂

∂+

∂

∂+

∂

∂

=Π

=

−−=∂

∂−=

π

ν

ρ

ττ

νπµµτ

For stepwise increase of velocity :

Integration over pipe section!

Convolution

∫∞

−

−

∂

∂−==

∂

∂=

0),(

0

),(2

0

0

|122

|),(

ξξπ

ν

ρ

τ

ρ

ξξ

ξ

ξ

dx

uc

RR

f

dx

utu

ctx

wx

ctx

)Pr

11(

−+

γFactor to include effect of heat transfer.

Chester (1964)

• Neglect change in average entropy

• Use non-linear simple wave theory

• Estimate friction for thin boundary layers

from linear theory

Thin boundary layer approach fails

for Sondhaus tube (Rott 1986)

Resonance of a closed pipe driven

by a piston

• Acoustics:

[ ]

)2exp(1

ˆ)2exp(

0),('

)]exp(Re[),0('

)exp()]exp()exp([Re),('

ikL

uikLpp

tLu

tiutu

tiikxpikxptxu

z

z

−−=−=

=

=

−−=

−+

−+

ω

ω)

L

p’(L,t)

Amplitude limited at resonance by friction

Sub-harmonic excitation (Keller 1975; Althaus and Thomann 1987)

Tt /

Tt /

zuc

tLp)

00

),('

ρ1

0

0 0.5

shocks

Due to reflection at close wall, we have long wave propagation distances!

L

p’(L,t)

Sub-harmonic excitation (Keller 1975; Althaus and Thomann 1987)

Thermal effects(Rott 1974-1980, Swift 1992, Bailliet 2001)

• Change of average entropy due to

dissipation and heat conduction.

• Thermo-acoustical devices.

Turbulence in standing wave(Merkli and Thomann 1975)

• Flow due to shock

us

Unstable jet flow

Uniform deceleration by shock

Non-uniform flow

Open tube resonance (Disselhorst and van Wijngaarden 1980)

• Reflection at open pipe termination results

in an inversion of acoustic wave

• Wave steepening from piston to open end is

compensated on the way back

Convective non-linearity

Pipe termination

acu acu

Vortex shedding

play

No play

Thin walled clarinet has narrower tone holes

(Keefe 1983, Atig, Dalmont and Gilbert 2004)

Vortex shedding modes depending on initial

conditions! (Disselhorst 1978)

a b

a

b

Self-sustained oscillation

Sound source

Resonator

Rijke tube

pipe

Heated

grid

Flow

Heat transfer

induces dilatation

Pipe is acoustical mass-spring system

wq 'wq

dt

dV'∝

Rijke tube

pipe

Heated

grid

acudt

dV'∝

acuU '0 +

acu'

acp' dt

dVpP ac'=

Optimal position:

impedance matching

Pipe mode Rayleigh

Power:

Self-sustained oscillation

Sound source

Resonator

Prediction of linear theory

acu'

>< acP

dissipation

production

2'acu∝

2'acu∝

Saturation due to non-linearity

acu'

>< acP

dissipation

production

2'acu∝

2'acu∝

Steady amplitude

Saturation

Saturation due to backflow

0U

d

acu '

0Uacu '

du ac

>ω

'

Conclusions

• “Complex problems have simple, easy to

understand wrong answers”

(N.Rott, 1986)

Some references

• L. Landau and E. Lifchitz, Mécanique des Fluides, Editions MIR, Moscou (1971)

• P.A. Thompson, Compressible-fluid Dynamics, McGrawHill , NY (1972)

• A. Pierce, Acoustics, McGrawHill (1981)

• D. Sette (editor), Frontiers in Physical Acoustics, “Enrico Fermi”course XCIII North-Holland (1986)

• A.Krothapalli and C.A. Smith (editors), Recent Advances in Aeroacoustics, Springer-Verlag, NY (1986)

• R.T. Beyer, Nonlinear Acoustics, ASA, NY (1997)

• K.Naugolnykh and L. Ostrovsky, Nonlinear wave processes in acoustics, Cambridge University Press, UK (1998)

• M.F.Hamilton and D.T. Blackstock, Nonlinear Acoustics, Academic Press, NY (1998)

• B.O. Enflo and C. M. Hedberg (editors), Theory of Nonlinear Acoustics in Fluids, Kluwer Academic Pub., Dordrecht (2002)

• Y.Aurégan, A.Maurel, V.Pagneux, J.-F. Pinton(Editors), Sound-Flow Interactions, Springer Verlag, Berlin (2002)

Self-similar solution viscous

boundary layer

∫ −−=

−=

=

=

x

cy

dc

uf

ff

x

cy

fcu

ν

ηη

π

η

νη

η

0

0

2

0

2

0

0

1)4

exp(

'2

1"

)(/