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Jiaqi_Wang
2020/11/23
Explore fluid/wave patterns in ducts
Outline
2020/11/23Explore fluid/wave patterns in ducts SJTU-SVN( )2
What is fluid/wave patterns? Why is it important?
Is all the emergence of a pattern predictable? Rayleigh-Bénard instability , Taylor–Couette instability, Duct modes in swirl flow
How to use the patterns in real turbomachinery? Some Applications.
1
2.1
2.2
My experience of “patterns”
2020/11/23Explore fluid/wave patterns in ducts SJTU-SVN( )3
https://www.newton.ac.uk/event/wht/seminars
Matthew Colbrook Sheehan OlverDavid Abrahams’s phd studentAnastasia Kisil & Matthew Priddin
Bringing pure and applied analysis together via the Wiener-Hopf technique, its generalisations and applications
黄迅
Similarity of patterns
2020/11/23Explore fluid/wave patterns in ducts SJTU-SVN( )4
My experience of “flow patterns”
2020/11/23Explore fluid/wave patterns in ducts SJTU-SVN( )5
https://fluids.ac.uk/sig/FlowInstability
“Flow instability, modelling and control”
AttractorVortex street
Peter Schmid
At same time, another workshop quietly launched in Cambridge
So, what is pattern?
2020/11/23Explore fluid/wave patterns in ducts SJTU-SVN( )6
-Wikipedia: A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner.
- In chao’s theory:
Koopman operator
- In fluid mechanism: Naiver-stoke equation
- In mathematician’s eyes :
- In engineer’s eyes : Resonance / Mode
System in thermodynamic equilibrium
2020/11/23Explore fluid/wave patterns in ducts SJTU-SVN( )7
Aspect solver for Earth's Convection
Rayleigh-Bénard (RB) Convection
Ra = 27⁄4 π4≈ 657.5
[1] Open source
Is all the emergence of a pattern predictable?
2020/11/23Explore fluid/wave patterns in ducts SJTU-SVN( )8
Consider a fluid between two parallel plates:
How large does V have to be for the flow to change the pattern?Early 20th century:All attempts to predict the answer failed.
Is all the emergence of a pattern predictable?
2020/11/23Explore fluid/wave patterns in ducts SJTU-SVN( )9
Experiment Taylor–Couette flow system
2020/11/23Explore fluid/wave patterns in ducts SJTU-SVN( )10
PIV
Explored Taylor–Couette flow system
2020/11/23Explore fluid/wave patterns in ducts SJTU-SVN( )11
[2]:Ruy Ibanez, Phys. Rev. Fluids,2016Taylor vortices wavy vortices chaotic flow
Increase Reynolds number
[3]: M. A. Fardin-2014
SimulationbyFEAToolMultiphysics
Explore fluid/wave patterns in ducts SJTU-SVN( )12
If patterns exit in more complex fluid dynamical system,Such like turbomachinery?Well, yes, some could be simplify,but more others are headache, confusing!
Contact me: www.deal-ii.com
Combustion instability in ducts
2020/11/23Explore fluid/wave patterns in ducts SJTU-SVN( )13
2
2
22
2
2
2
2)(111
trp
czpp
rrpr
rr ∂∂
=∂∂
+∂∂
+
∂∂
∂∂
θ
pressure mode superposition:
∑∑∞
=
∞
=
−−=0 0
)()cos()(m n
zktjmmnmmn
zemrkJAp ωϕθ2
2 2 22,z mnk k k k
cω
= − =轴向波数
( )0 0( )m mn
rm r amn r a
dJ k rud k r=
=
= ⇒ =Assuming that the wall is rigid, the normal velocity of the pipe wall is 0
Harmonic modes are governed by aHelmholtz equation
Axial wave number
Combustion rotating modes captured
2020/11/23Explore fluid/wave patterns in ducts SJTU-SVN( )14
[4]:S. Candel- Princeton summer school, June 2019
Compressor rotating stall
2020/11/23Explore fluid/wave patterns in ducts SJTU-SVN( )15
KuliteXCL-062
chord measurement layout of rotor
Pattern of rotating stall
2020/11/23Explore fluid/wave patterns in ducts SJTU-SVN( )16
2020/11/23Explore fluid/wave patterns in ducts SJTU-SVN( )17
Factor 1:Factor 2:Factor 3:Factor 4:Factor 5:Factor 6:Factor 7: Conical Duct
Factor 8: viscosity、turbulence
0u ( )r1 2R r R≤ ≤
u ( )rθ
0s ( )r
1 2( ) ( ) ( )y x r x y x≤ ≤
,t hZ Z
Realistic flow
Should be considered both for combustion and aeroacoustics
Wave pattern in swirling flow
Realistic flow: Eigenvalue
18
Linear Euler equations :
( )2
0 002 2
0 0
1 0D p U
v uDtc rc
θρρ+ + ∇ ⋅ =
00 ) 0
D u dUx pvDt dr x
ρ ∂+ + =
∂(
20
02
0D v U w U pDt r r r
θ θρ ρ ∂− − + =
∂( )
( )00
1 0D w v d prUDt r dr rθρ
θ∂ + + = ∂
0 0 0D s ds
vDt dr
+ =
Wave domain equation
2
2 2 2 20 0 0 0 0
2 2
20 0 0
0
20 0 0
20
ˆ ˆ 1 1
ˆ 2 1 ˆ
0
0
1
ˆ 1
0 0
x x
x x x p xx
x x x
x x x
U U UdUx d mi i idr r dr rc c rc c
U U UdiU rU U dr rc UU rc
U dU imiU r dr U r U
dU U U Udr rc
θ
θ θ θ
θ θ
θ
ζ ζ ζζ ζ ζ ρ ζ
ρ ρ
ρ
ρ ρ ρζ ζ
Ω Ω− + + + −
Ω− − −
Ω + −
Ω− − + −
Α =
02
0
0
ˆ
1 ˆ
0
0 0 0
x x
x
m U Ud idr r c
dsi
U dr
ρζ ζ ζ
Ω
−
−
Ω
UVWPS
=
X
kiλ = −
( ) ( ) ( ) ( ) ( ) ( ) n
, , , , , , , , , , , ikx in i tu v w p s r x t U r V r W r P r S r e dke e dθ ωθ ω−= ∑∫ ∫
( ) ( ) ( ) x
nU rr kU r
rθωΩ = − −
2 20
ˆ , 1 xnU
U crθω ζΩ = − = −
( )Explore fluid/wave patterns in ducts SJTU-SVN
2020/11/23
2020/11/23
Numerical method
19
Chebyshev Point
cos , 0, , Njjr j
Nπ
= =
( ) ( ) ( )0
N
jj
jf r f r rg=
= ∑
( ) ( )0
, 0, ,j
j
Ni
ij idf r
D f rdr
N=
= =∑
2
2
; 0, /2 0, ,
;
( 1)
2 sin ( )sin ( )
1, /2 1, ,
2 1
2 2
cos( )
2sin (
)
06
i ji
j
ij
i j i N j N
i j i
c
c i j i jN N
iND iN
N j N
N i j
π π
π
π
+−
+ − +
−=
≠ = =
= = =
+= =
;
;
, / 2 1, 0, ,N i N jD i N N j N− −
− =
+ = ;
Ingard-Myers boundary condition:
( )( )
( ) ( )( ) ( )
ˆ 0h
x x
V h h P hZ kP h
U h U hω Ω
+ − =
( )( )
( ) ( )( ) ( )1
1 1 11
10
1
ˆ
x x
V PZ kP
U Uω Ω
− + =
( )Explore fluid/wave patterns in ducts SJTU-SVN 2020/11/23
Explore the pattern from eigenvalue
20
h=0.6,w=25,n=15,Ux=0.5 solid wall
The eigenvalue is the axial wave number, which reflects the acoustic propagation characteristics of the pipe.
(near/pure) acoustic mode
Vortex-dominant (near/pure) convective mode
Swirling changes the cutoff and propagation characteristics of pipe acoustic propagation
Swirl is one of the acoustic resonance factors in the area of the inlet chute (Copper&Peake 2010 JFM)
( )( )
n
, , , ikx in i t
p r x t
P r e dke e dθ ω
θ
ω−= ∑∫ ∫
𝑛𝑛𝑛𝑛 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠m /xw kU U rθ− + +
convected mode
1 1inf sup
(r) (r)h r h rx x
w wkU U< < < <
≤ ≤
1 1
- (r) - (r)inf sup
(r) (r)h r h rx x
w nU w nUk
U Uθ θ
< < < <≤ ≤
𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
( ) ( )2/ 1x xreal k wU U= − −
𝑛𝑛𝑛𝑛 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 cut-offline
( )Explore fluid/wave patterns in ducts SJTU-SVN 2020/11/23
Application: Acoustic Analogy
21
CFDDuct AcousticSound
Radiation
Tonal Blade Noise
Modeling
2020/11/23( )Explore fluid/wave patterns in ducts SJTU-SVN
22
Acoustic Analogy: realistic flowLighthill eq.:
Condition:• medium(static、uniform);• Not observed points located
in the potential flow field;
• M<0.3。
Condtion:• considering the effects of
swirl, non-uniform entropy, shear flow, soft wall boundary conditions, etc.;( ) M MF p S=%
2 22 220 0 0 0
2 2 2 2 2 20 0
0 0 0 0
1 1 1: ( ) ( Y ( ) ) 0
[2 2 ( ) ]
Mx
x
D D U DF U
r Dt x Dtc Dt x r rcD D D U D
UDt r Dt x Dt r Dt
θθ
θθ
ρθ ρθ
θ
′∂ ∂ ∂ ∂′= − − ℜ + − − + − ℜ∂ ∂∂ ∂
∂ ∂ ∂′ ′ ′+ℜ − + + ℑ∂ ∂ ∂
Τ
Τ Τ
Posson&Peake eq.:(JFM 2012)Considering the influence of the axial shear and the rotating base flow in the pipeline, the equation is the form of the pressure disturbance under the action of the sixth-order linear operator.
20
02 2 2( ) S( ) S( )0 0 0
1 1 1(x, t) (y) (y)c c c
T T T
ij ii j iT v T T
D GG GT dyd f dS d V n dS dy y yτ τ τ
ρ τ τ ρ ττ− − −
∂ ∂′ ′ ′= + +∂ ∂ ∂ ∂∫ ∫ ∫ ∫ ∫ ∫
r r r rQuadrupole Dipole Monopole
Through the eigenfunction method, the Green's formula for satisfying the boundary conditions of the pipe wall for an infinitely long pipe can be derived.
*,n 2 3 ,n 2 3
, ,
,01 1 1 12 2+
,-
(y , y ) (x , x )(y, | x, t)
4
exp ( t) ( )
m m
m n m n
n m
n m
iG
kMki w y x y x
dwk
τπ
τβ β∞
∞
Ψ Ψ= ×
Γ
− + − + −
∑
∫
r r
2020/11/23( )Explore fluid/wave patterns in ducts SJTU-SVN
Posson&Peake’s Green function
23
( ) 0 00 0(x, t | x , t ) (x-x ) (t-t )M GF δ δ=r rr r
FFT,expand into cylinder coordination,
( )iwt00
00 0 0
( - ) (x-x ) ( - ) ( -(x | x ) e e ) eM iwt iw
wtr r
F x xr
Gδ
δ δ δ θ θ− − −==r rr r
( )M0 0 0 0 0x, t (x, t | x , t( ) ) S (x , t)G d dp x t= ∫
r r r r
( ) 00 0 0 0 0 0x, t ( |( ) , , , , ) S( , ), iwtwp r x r x r xG dx eθ θ θ −= ∫
r
Gw FFT to Wavenumber domain:
0 0( ) ( )00 02
1(x | x ) ( | ; , )4
in ik x xw n
nG e G r r w k e dkθ θ
π
∞− −
=−∞
= ∑ ∫¡
r r 0( )0 0(x, r | x , r )in m
w nn Kn
G e Gθ θ
±
∞−
=−∞
≈ ∑ ∑
clue Derivation wavenumber domain Green function Gn
Satisfy B.C.Accumulate into time domain
eigenvalue solution
2020/11/23( )Explore fluid/wave patterns in ducts SJTU-SVN
Monopole
24
Sound lining regular mode and reduce noiseThe swirl changes the number of convection waves to enhance a certain order of modes
NoSwirl
Swirl
Symmetrical distribution without swirling mode; rapid attenuation after the absolute value of the circumferential mode exceeds 25The swirl energy ratio is shifted to the negative direction
2020/11/23( )Explore fluid/wave patterns in ducts SJTU-SVN
Dipole
25 2020/11/23( )Explore fluid/wave patterns in ducts SJTU-SVN
Rotating
26 2020/11/23( )Explore fluid/wave patterns in ducts SJTU-SVN
Fan noise
27 2020/11/23( )Explore fluid/wave patterns in ducts SJTU-SVN
Acoustic Analogy: Tonal Noise
28
Extract unsteady CFD data on the surface
of the blade;
Calculate unsteady aerodynamic forces
on the surface of the blade;
Calculate the sound source item
(determine the rotating coordinate
system, position, etc.);
Solve the acoustic response equations at
the BPF and at each Harmonic position;
Calculate inlet sound field and post
processing
2020/11/23( )Explore fluid/wave patterns in ducts SJTU-SVN
29
Application: stall warning
2020/11/23( )Explore fluid/wave patterns in ducts SJTU-SVN
30
Application: Dynamic mode decomposition
2020/11/23( )Explore fluid/wave patterns in ducts SJTU-SVN
2019-11-1031
DMD:
特征值
特征向量
𝜆𝜆𝑘𝑘
[ , ]=eig( )k Aλ Φ %
𝛷𝛷
Application: Dynamic mode decomposition
Explore fluid/wave patterns in ducts SJTU-SVN( )
32
𝑠𝑠𝑘𝑘 = 𝑠𝑠𝑛𝑛(𝜆𝜆𝑘𝑘 ⁄) 𝑇𝑇
*kw w m SSF= ±Coordinate system conversion
(Positive and negative corresponding modal rotationPositive and negative directions))
Application: Dynamic mode decomposition
( )Explore fluid/wave patterns in ducts SJTU-SVN
2019-11-1033
Application: Dynamic mode decomposition
RS
RI
Explore fluid/wave patterns in ducts SJTU-SVN( )
2019-11-1034
Application: Mode detection2
20
1 D( ) 0Dt
p pc
− ∆ =
( ) [ ( ) ( )]mn mn mn m mn mn m mnE r C J r Q Y rκ κ κ= +
i(2 )
0( , , , ) ( ) e mnft m x
f mnf mn mnm n
p x r t A E r π θ ζθ κ+∞ +∞
+ −
=−∞ =
= ∑ ∑
i( ) e mf mf
mp a θθ
+∞−
=−∞
= ∑
1
1 ( ) k
Kim
mf f kk
a p eK
θθ=
= ∑
2 im -im2
1 1
im -im im -im2 2
1 1 1 1
1 1 ( )e ( )e2 2
1 1e ( ) ( )* e = e C e
k l
k l k l
K K
mf mf f k f lk k
K K K K
f k f l klk k k k
a p pK
p pK K
θ θ
θ θ θ θ
θ θ
θ θ
= =
= = = =
Γ = = =∑∑
∑∑ ∑∑
RMS
Explore fluid/wave patterns in ducts SJTU-SVN( )
2019-11-1035
Application: Mode detection5000rpm
6000rpm
m nB kV= ±B=29,V=32
10000rpm
-1 1
Explore fluid/wave patterns in ducts SJTU-SVN( )
谢 谢!感谢聆听!