Explore fluid/wave patterns in ducts

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Jiaqi_Wang

2020/11/23

Explore fluid/wave patterns in ducts

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Outline

2020/11/23

Explore 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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My experience of “patterns”

https://www.newton.ac.uk/event/wht/seminars

Matthew Colbrook Sheehan Olver David Abrahams’s phd student

Anastasia Kisil& Matthew Priddin

Bringing pure and applied analysis together via the Wiener-Hopf technique, its generalisations and

applications

黄迅

(4)

Similarity of patterns

2020/11/23

(5)

My experience of “flow patterns”

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

“Flow instability, modelling and control”

Attractor Vortex street

Peter Schmid

At same time, another workshop quietly

launched in Cambridge

(6)

So, what is pattern?

2020/11/23

-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 mechanis m: Naiver-stoke equation - In mathematician’s eyes :

- In engineer’s eyes : Resonance / Mode

(7)

System in thermodynamic equilibrium

Aspect solver for Earth's Convection

Rayleigh-Bénard (RB) Convection

Ra = 27⁄4 π4

≈ 657.5

[1] Open source

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Is all the emergence of a pattern predictable?

2020/11/23

 Consider a fluid between two parallel plates:

How large does V have to be for the flow to change the pattern?

Early 20

^th

century:

All attempts to predict the answer failed .

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Is all the emergence of a pattern predictable?

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Experiment Taylor–Couette flow system

2020/11/23

PIV

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Explored Taylor–Couette flow system

[2]:Ruy Ibanez, Phys. Rev. Fluids,2016

Taylor vortices wavy vortices chaotic flow Increase Reynolds number

[3]: M. A. Fardin-2014

Simulation byFEATool Multiphysics

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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

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Combustion instability in ducts

2 2 2 2

2 2

) ( 1

1 1

t rp c

z p p

r r

r p r

r ∂

= ∂

∂ + ∂

 

 





∂

θ

 pressure mode superposition :

∑∑ ^∞

=

∞

=

− −

=

0 0

)

) (

cos(

) (

m n

z k t m j

mn m

mn J k r m e

^z

A

p θ ϕ ^ω

2 2 2 2

,

2

z mn

k k k k

c

= − = ω

轴向波数

( )

0 0

( )

m mn

rm r a

mn r a

dJ k r

u

⁼

= ⇒ d k r

₌

= Assuming that the wall is rigid,

the normal velocity of the pipe wall is 0

Harmonic modes are governed by a Helmholtz equation

Axial wave number

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Combustion rotating modes captured

2020/11/23

[4]:S. Candel- Princeton

summer school, June 2019

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Compressor rotating stall

KuliteXCL-062

chord measurement layout

of rotor

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Pattern of rotating stall

2020/11/23

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Factor 1：

Factor 2：

Factor 3：

Factor 4：

Factor 5：

Factor 6：

Factor 7： Conical Duct

Factor 8： viscosity、turbulence

u ( ) ^R

0

¹ r ^{r R} ²

≤ ≤ u ( ) _θ r

s ( )

0

r

1

( ) ( )

2

( )

y x ≤ r x ≤ y x

t

,

h

Z Z

Realistic flow

 Should be considered both for combustion and aeroacoustics

Wave pattern in swirling flow

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Realistic flow: ^Eigenvalue

18

Linear Euler equations : ( )

0 0 2

2 2 0

0 0

1 D p U v u 0

c Dt rc

ρ

θ

ρ

+ + ∇ ⋅ =

0

D u

0

v dUx ) p 0

Dt dr x

ρ + + ^∂ =

（ ∂

0 2 0

2 0

D v U w U p

Dt r^θ r^θ r

ρ − − ρ+^∂ =

（） ∂

( )

0 D w v d0 rU 1 p 0

Dt r dr ^θ r

ρ θ

 + + ∂ =

  ∂

 

0 0

0 D s ds v Dt + dr =

Wave domain equation

2

2 2 2 2

0 0 0 0 0

2 2

0 0 20

0

0 0 2 0

20

ˆ 1 1 ˆ

ˆ 2 1

ˆ

0

1

ˆ 1

0 0

x x

x x x x p x

x x x

U i U dUx U d m i i

dr r dr r

c c rc c

U d U U

iU rU U dr U rc rc U

U dU i im

U r dr U r U

dU U U U

dr c r

θ

θ θ θ

θ θ

θ

ζ ζ ζ

ζ ζ ζ ρ ζ

ρ ρ

ρ

ρ ρ ρ

ζ ζ

 

Ω − + + + − Ω

 

− Ω − −

 +  − Ω

 

 

   

− Ω  − +  −

   

 

Α =

0

20 0

ˆ

1 ˆ

0

0 0 0

x x

x

m U U

d i

dr r c

ds i

U dr

ρ

ζ ζ ζ

 

 

 Ω

−

 

 

 Ω 

U V W

P S

  

  

=  

  

  X

ki λ = −

{ }( ) { ( ) ( ) ( ) ( ) ( ) }

n

, , , , , , , , , , ,

^ikx ⁱⁿ ^{i t}

u v w p s r x θ t = ∫ ∫ ∑ U r V r W r P r S r e dke e

^θ ⁻ ^ω

d ω

( )

( ) ( )

x

r kU r nU r

r

ω

θ

Ω = − −

2 2

ˆ nU , 1 U c_x 0

r^θ

ω ζ

Ω = − = −

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Explore fluid/wave patterns in ducts SJTU-SVN

2020/11/23

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Numerical method

Chebyshev Point

cos , 0, , N

j

r j

N

= π = 

( ) ( ) ^{( )}

0 N

j j j

f r f r g r

=

= ∑

( ) ( )

0

, 0, ,

j j

i N

ij

i

df r D f r

dr N

=

= ∑ = ^

2 2

; 0, /2 0, ,

; ( 1)

2 sin ( )sin ( )

1, /2 1, ,

2 1

2 2

cos( ) 2sin (

)

6 0

i i j

j

ij

i j i N j N

i j i c

c i j i j

N N

i

D Ni

N

N j N

N i j

π π

− +

+ − +

−

=

≠ = =

= = =

+ = =

 

；

D_{N i N j}−, − i N/ 2 1, N j 0, ,N







− =





 +  ； = 

Ingard-Myers boundary condition：

( ) ( ) ( ) ( )

( ) ( )

ˆ 0

h

x x

V h h P h

Z kP h

U h U h

ω Ω

+ − =

( ) ( ) ( ) ( )

( ) ( )

1

1 1 1

1 1 0

1 ˆ

x x

V P

Z kP

U U

ω Ω

− + =

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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 d

^θ ^ω

θ

ω

= ∫ ∫ ∑

−

𝑛𝑛𝑛𝑛 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠

m / w kU

x

U r

_θ

− + +

 convected mode

1 1

inf sup

(r) (r)

h r x h r x

w k w

U U

< < ≤ ≤ < <

1 1

- (r) - (r)

inf sup

(r) (r)

h r x h r x

w nU k w nU

U U

θ θ

< < ≤ ≤ < <

𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠

( )

x

^{/ 1} (

x²

)

real k = − wU − U

 𝑛𝑛𝑛𝑛 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 cut-offline

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Explore fluid/wave patterns in ducts SJTU-SVN ^2020/11/23

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Application: Acoustic Analogy

Duct Acoustic CFD Sound

Radiation

Tonal

Blade

Noise

Modeling

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22

Acoustic Analogy: realistic flow

Lighthill 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 M

F p % = S

²⁰ ⁰²² ²² ² ²² ² ⁰ ⁰²² ⁰ ⁰

0 0 0 0

1 1 1

: ( ) ( Y ( ) )

0

[2 2 ( ) ]

M x

x

D D U D

F U

r Dt x Dt

c Dt x r rc

D D U D U D

Dt r Dt x Dt r Dt

θ θ

ρ

θ ρ

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.

2 0

2 2 2 0

( ) S( ) S( )

0 0 0

1 1 1

(x, t) (y) (y)

c c c

T T T

ij i

i j i

T v T T

G T dyd G f dS d V n D G dS d

y y y

τ τ τ

ρ τ τ ρ τ

− − −

τ

∂ ∂

′ = ′ + + ′

∂ ∂ ∂ ∂

∫ ∫ ∫ ∫ ∫ ∫

r ^Quadrupole r Dipole Monopole r r

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

, ,

0 ,

1 1 1 1

2 2

+ - ,

(y , y ) (x , x ) (y, | x, t)

4

exp ( t) ( )

m m

m n m n

n m

G i

Mk k

i w y x y x

k dw

τ π

τ β β

∞

Ψ Ψ

= ×

Γ

  

 − + − + − 

  

 

 



∑

∫

r r

2020/11/23

（）

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Posson&Peake’s Green function

( (x, t | x , t ) (x-x ) (t-t )

₀ ₀

)

₀ ₀

M

G

F r r = δ r r δ

FFT，expand into cylinder coordination，

(

00 ^iwt

)

0

0 0 0

( - )

(x-x ) ( - ) ( -

(x | x ) e e ) e

M iwt iw

w r r t

F x x

G

δ

r

δ δ δ θ θ

− = r r − = −

r r

(

0 0 ^M 0

)

0 0

x, t (x, t | x , t

( ) G )S (x , t) d d

p ^r = ∫ ^r ^r ^r x t

(

₀ ₀ ₀ ₀ ₀ ₀

)

₀

x, t ( |

( )

_w

, , , , )S( , , )

^iwt

p ^r = ∫ G r x θ r x θ r x θ dx e

⁻

Gw FFT to Wavenumber domain：

0 0

( ) ( )

00 2 0

(x | x ) 1 ( | ; , )

4

in ik x x

w n

n

G e

^{θ θ}

G r r w k e dk

π

∞ − −

=−∞

= ∑ ∫

¡

r r

( 0)

(x,r | x ,r )

0 0

in m

w n

n Kn

G e

^{θ θ}

G

±

∞ −

=−∞

≈ ∑ ∑

clue Derivation wavenumber domain Green

function Gn

Satisfy B.C.

Accumulate into time

domain

eigenvalue solution

(24)

Monopole

24

Sound lining

regular mode and reduce noise

The swirl changes the number of convection waves to enhance a

certain order of modes

No Swirl

Swirl

Symmetrical

distribution without swirling mode;

rapid attenuation after the absolute value of the

circumferential mode exceeds 25 The swirl energy ratio is shifted to the negative direction

2020/11/23

（）

(25)

Dipole

(26)

Rotating

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

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Fan noise

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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

（）

(29)

Application: stall warning

(30)

30

Application: Dynamic mode decomposition

2020/11/23

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DMD：

特征值

特征向量

𝜆𝜆 _𝑘𝑘

[ , ]=eig( ) λ

_k

Φ A %

𝛷𝛷

Application: Dynamic mode decomposition

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32

𝑠𝑠 _𝑘𝑘 = 𝑠𝑠𝑛𝑛(𝜆𝜆 _𝑘𝑘 ) 𝑇𝑇 ⁄

k *

w w = ± m SSF

Coordinate system conversion

（Positive and negative

corresponding modal rotation Positive and negative directions)）

Application: Dynamic mode decomposition

（）

(33)

Application: Dynamic mode decomposition

RS

RI

(34)

2019-11-10

34

Application: Mode detection

2 20

1 D( ) 0

Dt p p

c − ∆ =

( ) [ ( ) ( )]

mn mn mn m mn mn m mn

E κ r =C J κ r Q Y+ κ r

i(2 )

0

( , , , ) ( )e ^{ft m} ^mn^x

f mnf mn mn

m n

p x r θ t ^+∞ ^+∞ A E κ r ^π ⁺ ^{θ ζ}⁻

=−∞ =

=

∑ ∑

( ) e

i^m

f mf

m

p θ

^+∞

a

⁻ ^θ

=−∞

= ∑

1

1 ^K ( ) ^im^k

mf f k

k

a p e

K

θ θ

=

∑

2 im -im

2 1 1

im -im im -im

2 2

1 1 1 1

1 1 ( )e ( )e

2 2

1 e ( ) ( )* e = 1 e C e

k l

k l k l

K K

mf mf f k f l

k k

K K K K

f k f l kl

k k k k

a p p

K p p

K K

θ θ

θ θ θ θ

θ θ

= =

= = = =

Γ = =

∑∑

=

∑∑ ∑∑

RMS

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

(35)

Application: Mode detection

5000rpm

6000rpm m nB kV = ±

B=29，

V=32 10000rpm

-1 1

(36)

Explore fluid/wave patterns in ducts

Jiaqi_Wang