On time domain methods for Computational Aeroacoustics Fang Q. Hu - - PowerPoint PPT Presentation

SLIDE 1

On time domain methods for Computational Aeroacoustics

Fang Q. Hu and Ibrahim Kocaogul Old Dominion University, Norfolk, Virginia Xiaodong Li, Xiaoyan Li and Min Jiang Beihang University, Beijing 100191, China

SLIDE 2

Noise prediction by linear acoustic wave propagation

FW−H equation Kirchhoff integral

Noise source modelling + Noise propagation

Linearized Euler Equations Green’s function ......

SLIDE 3

Time Domain Wave Packet (TDWP) method

sinusoidal wave wave packet (single frequency) (broadband and mulch-frequency) Proposed broadband acoustic test pulse function for source: Ψ(

t) = ∆ t sin(ω t)

π

t e( ln

• 0. 01)( t/M∆t)

2, | t| ≤ M∆ t

SLIDE 4

Advantages of Time Domain Wave Packet (TDWP) method

◮ One computation for all frequencies (within numerical

resolution) for linear problems

◮ Ability to synthesize broadband noise sources ◮ Acoustic source has a short time duration, so computation is

more efficient than driving a time domain calculation to a time periodic state

◮ Separation of acoustic and hydrodynamic instability waves

becomes possible

◮ Long numerical transient state is avoided

SLIDE 5

Application Examples

• 1. Sound propagation through shear flows
• 2. Vortical gust-blade interaction
• 3. Duct sound radiation problem

SLIDE 6

• 1. Sound source in a jet flow (a CAA Benchmark problem)

x=100 Jet Flow y=15 y=50

Single frequency source function:

S( x, y, t) = sin(Ω t) e−(ln 2)(B x x 2+B y y 2)

SLIDE 7

Instability wave in a shear flow

x y

• 50

50 100 150

• 60
• 40
• 20

20 40 60

SLIDE 8

Instability wave in duct radiation computation

SLIDE 9

Time Domain Wave Packet (TDWP) method

Separation of acoustic and instability waves:

instability wave

Shear layer

1 2

t t t3

Acoustic and instability waves travel at different speeds. An acoustic wave packet has a short time duration, it will be separated from the instability wave in time domain calculation, here

t 1 < t 2 < t 3.

SLIDE 10

Time Domain Wave Packet approach

x=100 Jet Flow y=15 y=50

Single frequency source function:

S( x, y, t) = sin(Ω t) e−(ln 2)(B x x 2+B y y 2)

TDWP source function:

S( x, y, t) = Ψ( t) e−(ln 2)(B x x 2+B y y 2)

SLIDE 11

without suppression with suppression

SLIDE 12

Frequency domain solution recovered by FFT

(Symbol: analytical; Line: computation)

y = 15 y = 50 x = 100

SLIDE 13

• 2. Vortical gust-blade interaction

Incident Vortical Gust :

(U ,V )

u g = − V β

α

• s(α

x + β y − ω t) v g = V

• s(α

x + β y − ω t)

SLIDE 14

Vortical gust imposition in TDWP

vortical wave packet

θ

u g( x, y, t) = −BΨ( t − Ax − By) v g( x, y, t) = AΨ( t − Ax − By) A =

• s(θ)

¯

u

• s(θ)+ ¯

v sin(θ), B = sin(θ)

¯

u

• s(θ)+ ¯

v sin(θ) =

⇒ ∂

u g

∂

x + ∂ v g

∂

y =

SLIDE 15

Example of time domain solution

(v-velocity and pressure,¯

u 0 =

• 0. 45)

SLIDE 16

Frequency domain solution by FFT (u-velocity and pressure)

SLIDE 17

Frequency domain solution (pressure magnitudes)

SLIDE 18

• 3. Duct mode imposition in TDWP

P M L

Source plane

Pressure equation at in-flow region:

∂

p

∂

t + ¯ u

∂

p

∂

x +γ¯ p

∂

u

∂

x + ∂ u

∂

y + ∂ u

∂

z

• = φ

mn( y, z)Ψ( t) e−σ x 2

SLIDE 19

Example of time domain solution

SLIDE 20

Duct mode wave packet—exact solution

p( x, y, z, t) = φ mn( y, z)ˆ P( x, t)

ˆ

P( x, t) = α 2 2

ˆ

X

− X

ˆ

T ′

− T

J

• ¯

λ

mn

• ( t −¯

t′) 2 − ¯ x 2

• D

Dt

• Ψ(

t′) e−σ x′ 2 dx′ dt′

ˆ

P( x,ω) =

              

α

2 2

´

X

− X

´

T ′

− T

e−¯ x√

¯ λ

2 mn−ω 2

√¯

λ

2 mn−ω 2 e iω¯ t′ D Dt

• Ψ(

t′) e−σ x′ 2 dx′ dt′ 0 < ω < ¯

λ

mn (cut-off)

α

2 2

´

X

− X

´

T ′

− T

ie i ¯ x√

ω

2−¯

λ

2 mn

√

ω

2−¯

λ

2 mn e iω¯ t′ D Dt

• Ψ(

t′) e−σ x′ 2 dx′ dt′ 0 < ¯

λ

mn < ω (cut-on)

• α =
• 1−

M 2, β = M 1− M 2 , ¯ t′ = t′ −β( x − x′), ¯ x = | x − x′|/α 2

SLIDE 21

Time history (line: numerical, symbol:theoretical)

SLIDE 22

Frequency domain solution

ω

1 = 1. 6 :

ω

2 = 1. 9 :

ω

3 = 2. 3 :

ω

4 = 2. 6 :

SLIDE 23

NASA/GE Fan Noise Source Diagnostic Test

SLIDE 24

Aft fan exhaust radiation problem

Pressure equation at in-flow region:

∂

p

∂

t + ¯ u

∂

p

∂

x +γ¯ p

∂

u

∂

x + ∂ u

∂

y + ∂ u

∂

z

• = φ

mn( y, z)Ψ( t) e−σ(x− x 0) 2

SLIDE 25

Aft fan exhaust radiation problem

SLIDE 26

Aft fan exhaust radiation problem

SLIDE 27

Aft fan exhaust radiation problem

SLIDE 28

Aft fan exhaust radiation problem

SLIDE 29

Pressure field obtained by FFT at 2BPF, 61.7% design speed

Mode (-10,2) introduced at source plane inside duct

SLIDE 30

Pressure field obtained by FFT at 3BPF, 61.7% design speed

Mode (-10,2) introduced at source plane inside duct

SLIDE 31

Pressure field obtained by FFT at 4BPF, 61.7% design speed

Mode (-10,2) introduced at source plane inside duct

SLIDE 32

Far-field modal transfer function, forward solution

51 50 49 45 40 35

Forward Radiation, mode (-10,0)

SLIDE 33

Far-field modal transfer function, forward solution

51 50 49 45 40

Forward Radiation, mode (-10,1)

SLIDE 34

Far-field modal transfer function, forward solution

51 50 49 47 45 35

Forward Radiation, mode (-10,2)

SLIDE 35

Far-field modal transfer function, forward solution

51 50 49 45 40 35

Forward Radiation, mode (-10,3)

SLIDE 36

Far-field modal transfer function, forward solution

Forward Radiation, mode (-10,4)

SLIDE 37

Far-field modal transfer function (2BPF) ˆ

P mn = P mn A mn

51 50 49 47 45 40 35

Forward Radiation

(−10,1) (−10,2) (−10,3) (−10,4) (−10,0)

SLIDE 38

Reciprocity Condition (forward and adjoint problems)

nm

ω

nm

A p (r’, ) ~ p(r’,t) Amn ~

P mn(r′,ω) A mn

= α

mn

˜

A∗ mn

˜

P(r′,ω)

• α

mn =

ˆ

D

˜ φ

∗

mn Aφ mn dS = 4π

∂ω ∂

k

• mn

ˆ

R r H

φ

2 mn( r) rdr

SLIDE 39

Adjoint solution, time domain

SLIDE 40

Adjoint solution, time domain

SLIDE 41

Adjoint solution, time domain

SLIDE 42

Adjoint solution, time domain

SLIDE 43

Adjoint solution, frequency domain at 2BPF

SLIDE 44

Adjoint solution, frequency domain at BPF

SLIDE 45

Adjoint solution, frequency domain at 3BPF

SLIDE 46

Reciprocity condition (2BPF)

P mn A mn = α mn

˜

A∗ mn

˜

P

51 50 49 47 45 40 35

Forward Radiation

45

Adjoint Solution

~

(−10,1) (−10,2) (−10,3) (−10,4) (−10,0)

SLIDE 47

Reciprocity condition (2BPF)

P mn A mn = α mn

˜

A∗ mn

˜

P

51 50 49 47 45 40 35

Forward Radiation

48

Adjoint Solution

~

(−10,1) (−10,2) (−10,3) (−10,4) (−10,0)

SLIDE 48

Reciprocity condition (2BPF)

P mn A mn = α mn

˜

A∗ mn

˜

P

51 50 49 47 45 40 35

Forward Radiation

49

Adjoint Solution

~

(−10,1) (−10,2) (−10,3) (−10,4) (−10,0)

SLIDE 49

Comparison of modal transfer function

P mn A mn = α mn

˜

A∗ mn

˜

P

51 50 49 47 45 40 35

Forward Radiation Adjoint Solution

~

(−10,1) (−10,2) (−10,3) (−10,4) (−10,0)

SLIDE 50

Modal detection 1

• Far-field pressure

p( r,ω) = ∑ m,n A mn(ω)ˆ P mn( r,ω) A mn = A(r) mn + iA(i) mn = Amplitude

ˆ

P mn = Modal transfer function

• Assume duct modes have no interference:

|p(

r,ω)| 2 = ∑ m,n

|A

mn(ω)| 2

• ˆ

P mn( r,ω)

• 2

(1)

• Minimization:

51

∑

i=38

• ∑

m,n

|A

mn| 2

• ˆ

P mn( r i,ω)

• 2

−

P 2 i (ω)

• 2

=

MIN

(2)

SLIDE 51

Far-field SPL

measurements computation

SLIDE 52

Summary

◮ Time Domain Wave Packet formulation can be used to

eliminate the initial long transient state that is often required in single frequency formulation

◮ Computational time is reduced due to shortened time

duration of the wave packet; the wave packet method is preferred for linear propagation problems even if only solutions at a few frequencies are of interest

◮ Solution of the adjoint problem provides a useful tool for

verifying the numerical results of Euler equations