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Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion

Separating propagating and non-propagating dynamics in fluid-flow equations

Samuel Sinayoko,

• A. Agarwal and Z. Hu

University of Southampton Institute of Sound and Vibration Research

May 2009

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion

Introduction

How to define the physical sources of sound? Objectives

1

Derive an expression for the physical sources of sound.

2

Demonstrate that it is possible to separate the radiating and non-radiating parts of the flow.

3

Compute the physical sources of sound.

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion Goldstein’s theory Equations

Goldstein’s theory

Jet Filtered jet

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion Goldstein’s theory Equations

Goldstein’s theory

Jet Filtered jet

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion Goldstein’s theory Equations

Goldstein’s theory

Jet Filtered jet These sources should be close to the true sources of sound.

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion Goldstein’s theory Equations

Governing equation for fluctuating quantities

Flow filtering Lf = f (1) Flow decomposition f = f + f ′ (2) Conservation of mass ∂ρ ∂t + ∂ρvj ∂xj = 0, (3) ∂ρ ∂t + ∂ρvj ∂xj = 0. (4) Conservation of mass for fluctuating quantities ∂ρ′ ∂t + ∂(ρvj)′ ∂xj = 0.

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion Goldstein’s theory Equations

Governing equation for fluctuating quantities

Conservation of mass for fluctuating quantities ∂ρ′ ∂t + ∂(ρvj)′ ∂xj = 0. (5) Momentum conservation for fluctuating quantities ∂(ρvi)′ ∂t + ∂(ρvivj)′ ∂xj + ∂p′ ∂xi = ∂σ′

ij

∂xj . (6) Taking ∂(6)/∂xi − ∂(5)/∂t gives ∂2p′ ∂xixi − ∂2ρ′ ∂t2 + ∂2(ρvivj)′ ∂xi∂xj = ∂2σ′

ij

∂xi∂xj . (7)

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion Goldstein’s theory Equations

Governing equation for fluctuating quantities

Favre averaging, ˜ f = ρf/ρ, (8) Governing equation ∂2p′ ∂xi∂xi − ∂2ρ′ ∂t2 + ∂2 ∂xi∂xj (˜ vi ˜ vjρ′ + ρ˜ vjv′

i + ρ˜

viv′

j ) = ∂2σij′

∂xi∂xj + s (9) Source definition s = − ∂2 ∂xi∂xj

• Tij + ρv′

i v′ j + ˜

viρ′v′

j + ˜

vjρ′v′

i

• (10)

Tij = −ρ( vivj − ˜ vi ˜ vj). (11)

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion Problem description Filter defining properties Local filter Global filter

Problem description

Parallel flow

−10 −5 5 10 y, m −50 50 100 150 x, m 0.2 0.4 0.6 0.8

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion Problem description Filter defining properties Local filter Global filter

Problem description

Pressure field

−50 50 y, m −50 50 100 150 x, m

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion Problem description Filter defining properties Local filter Global filter

Defining properties

Fourier transform f(x, t) → F(k, ω) f(x, t) → F(k, ω) Non-radiating condition F(k, ω) = 0 for |k| = |ω| c∞ Additional requirement F(k, ω) = F(k, ω) for |k| = |ω| c∞

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion Problem description Filter defining properties Local filter Global filter

Local filter

Filter definition

D’Alembertian filter f(x, t) = 1 c2

∞

∂2 ∂t2 − ∇2

• f(x, t),

Frequency domain F(k, ω) =

• |k|2 − ω2

c2

∞

• F(k, ω)

⇒ F(k, ω) = 0 for |k| = |ω|

c∞

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion Problem description Filter defining properties Local filter Global filter

Local filter

Results

−50 50 y, m −50 50 100 150 x, m

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion Problem description Filter defining properties Local filter Global filter

Local filter

Results

−50 50 y, m −50 50 100 150 x, m

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion Problem description Filter defining properties Local filter Global filter

Global filter

Filter definition

Time-domain f = w ∗ f (12) Frequency-domain F = WF (13)

f(x, t) F(k, ω) Filter window if |k| = |ω|/c∞ 1

• therwise

F(k, ω) f(x, t) F .T. I.F.T.

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion Problem description Filter defining properties Local filter Global filter

Global filter

Filter definition

Gaussian filter W(k, ω) = exp

• −(kx − α)2

2σ2

• + exp
• −(kx + α)2

2σ2

• α = 0.68m−1,

σ = 0.1m−1.

kx W(k, ω) α −α 1 4σ

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion Problem description Filter defining properties Local filter Global filter

Global filter

Results

−50 50 y, m −50 50 100 150 x, m

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion Problem description Filter defining properties Local filter Global filter

Global filter

Results

−50 50 y, m −50 50 100 150 x, m

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion Problem description Filter defining properties Local filter Global filter

Global filter

Validation

Comparison with analytical result along profile y = 15m

−2.5×10−6 2.5×10−6 pressure −50 50 100 150 x, m

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion Flow description Filter design Sound sources

Flow description

Mean flow

−5 5 y 10 20 x 0.2 0.5 0.8 1

Mean flow excited at two frequencies: ω1 = 2.2, ω2 = 3.4, ∆ω = 1.2.

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion Flow description Filter design Sound sources

Flow description

Pressure field

3 6 9 12 15 y 10 20 x −5 5 ×10−6

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion Flow description Filter design Sound sources

Flow description

Frequency analysis

Hydrodynamic region

−100 −80 −60 −40 −20 Power, dB 1 2 3 4 5 6 ω ω1 ω2

Acoustic region

−160 −140 −120 −100 −80 Power, dB 1 2 3 4 5 6 ω ∆ω

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion Flow description Filter design Sound sources

Flow description

Frequency analysis

Hydrodynamic region

−100 −80 −60 −40 −20 Power, dB 1 2 3 4 5 6 ω ω1 ω2 ωco

Acoustic region

−160 −140 −120 −100 −80 Power, dB 1 2 3 4 5 6 ω ∆ω

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion Flow description Filter design Sound sources

Filter design

Definition

Tanh filter W(k, ω) = 1 2

• 1 + tanh

|k| − kco σ

• ,

kco = 1.3, σ = 0.2.

|k| W(k, ω) kco 1

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion Flow description Filter design Sound sources

Filter design

Validation

Pressure field p

3 6 9 12 15 y 10 20 x

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion Flow description Filter design Sound sources

Filter design

Validation

Filtered pressure p

3 6 9 12 15 y 10 20 x

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion Flow description Filter design Sound sources

Filter design

Validation

Fluctuating pressure p′

3 6 9 12 15 y 10 20 x

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion Flow description Filter design Sound sources

Sound sources

Using non-radiating filter

Sound source s

1 2 y 2 4 6 8 10 x

Spectrum at (4.0, 0.55)

−80 −60 −40 −20 Power, dB 0 1 2 3 4 5 6 ω ∆ω

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion Flow description Filter design Sound sources

Sound sources

Using time average filter

Sound source s

1 2 y 2 4 6 8 10 x

Spectrum at (5.5, 0.5)

−60 −40 −20 Power, dB 1 2 3 4 5 6 ω ∆ω

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion Flow description Filter design Sound sources

Sound sources

Evolution in time

(source)

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion

Conclusion and future work

Results Sound source definition Separation possible with convolution filters. Clearer physical interpretation of the sources. Future work Mixing-layer and a two-dimensional jet. Physical mechanism behind the sound sources.

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Non-radiating filter design Sources of sound in an axi-symmetric jet Conclusion

Acknowledgements

• S. Sinayoko, A. Agarwal, Z. Hu
• Sep. propagating & non-propagating dynamics in fluid-flow