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
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
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
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
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
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
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 ++=
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)
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
Vortex shedding
play
No play
Thin walled clarinet has narrower tone holes
(Keefe 1983, Atig, Dalmont and Gilbert 2004)
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:
Saturation due to non-linearity
acu'
>< acP
dissipation
production
2'acu∝
2'acu∝
Steady amplitude
Saturation
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)
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