research landscape of china

Jiaqi_Wang

2020/11/23

Explore fluid/wave patterns in ducts

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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

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What is fluid/wave patterns? Why is it important? How to explore? How to use the patterns?

My experience of “patterns”


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

http://www.damtp.cam.ac.uk/user/mjc249/home.html

https://wwwf.imperial.ac.uk/%7Esolver/

https://en.wikipedia.org/wiki/David_Abrahams_(mathematician)

http://www1.maths.leeds.ac.uk/%7Ekisilv/our-kids.html

http://www2.coe.pku.edu.cn/subpaget.asp?id=199

Similarity of patterns


My experience of “flow patterns”


https://fluids.ac.uk/sig/FlowInstability

“Flow instability, modelling and control”

AttractorVortex street

Peter Schmid

At same time, another workshop quietly launched in Cambridge

https://en.wikipedia.org/wiki/Attractor

https://en.wikipedia.org/wiki/Vortex_street

https://en.wikipedia.org/wiki/Vortex_street

https://www.imperial.ac.uk/people/peter.schmid

So, what is pattern?


-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


Aspect solver for Earth's Convection

Rayleigh-Bénard (RB) Convection

Ra = 27⁄4 π4≈ 657.5

[1] Open source

https://aspect.geodynamics.org/

https://zh.wikipedia.org/zh-hans/%E7%91%9E%E5%88%A9-%E8%B4%9D%E7%BA%B3%E5%BE%B7%E5%AF%B9%E6%B5%81

https://paleodyn.uni-bremen.de/study/FortranBook.pdf

Is all the emergence of a pattern predictable?


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?


Experiment Taylor–Couette flow system


PIV

https://www.researchgate.net/publication/225451342_Qualitative_relation_between_reflected_light_intensity_by_Kalliroscope_flakes_and_velocity_field_in_the_Couette-Taylor_flow_system/figures?lo=1

Explored Taylor–Couette flow system


[2]:Ruy Ibanez, Phys. Rev. Fluids,2016Taylor vortices wavy vortices chaotic flow

Increase Reynolds number

[3]: M. A. Fardin-2014

SimulationbyFEAToolMultiphysics

https://pubs.rsc.org/en/content/articlelanding/2014/sm/c3sm52828f#!divAbstract

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


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


[4]:S. Candel- Princeton summer school, June 2019

Compressor rotating stall


KuliteXCL-062

chord measurement layout of rotor

Pattern of rotating stall



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

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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

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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

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图中三种轴向速度参数下，其cut-on声模态个数依次递减；同时，cut-off所在的直线也逐渐右移。

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


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


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


Dipole

25 2020/11/23（）Explore fluid/wave patterns in ducts SJTU-SVN

Rotating


Fan noise


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


29

Application: stall warning


30

Application: Dynamic mode decomposition


2019-11-1031

DMD：

特征值

特征向量

𝜆𝜆𝑘𝑘

[ , ]=eig( )k Aλ Φ %

𝛷𝛷


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)）


（）Explore fluid/wave patterns in ducts SJTU-SVN

2019-11-1033


RS

RI


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


2019-11-1035

Application: Mode detection5000rpm

6000rpm

m nB kV= ±B=29，V=32

10000rpm

-1 1


谢谢！感谢聆听！

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