Separating propagating and non-propagating dynamics in fluid-flow - PowerPoint PPT Presentation

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 Derive an expression for the physical sources of sound. 1 Demonstrate that it is possible to separate the radiating 2 and non-radiating parts of the flow. Compute the physical sources of sound. 3 S. Sinayoko, A. Agarwal, Z. Hu Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Goldstein’s theory Non-radiating filter design Equations Sources of sound in an axi-symmetric jet Conclusion Goldstein’s theory Filtered jet Jet S. Sinayoko, A. Agarwal, Z. Hu Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Goldstein’s theory Non-radiating filter design Equations Sources of sound in an axi-symmetric jet Conclusion Goldstein’s theory Filtered jet Jet S. Sinayoko, A. Agarwal, Z. Hu Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Goldstein’s theory Non-radiating filter design Equations Sources of sound in an axi-symmetric jet Conclusion Goldstein’s theory Filtered jet 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 Goldstein’s theory Non-radiating filter design Equations Sources of sound in an axi-symmetric jet Conclusion Governing equation for fluctuating quantities Flow filtering Flow decomposition f = f + f ′ L f = f (1) (2) Conservation of mass ∂ t + ∂ρ v j ∂ρ = 0 , (3) ∂ x j ∂ t + ∂ρ v j ∂ρ = 0 . (4) ∂ x j Conservation of mass for fluctuating quantities ∂ t + ∂ ( ρ v j ) ′ ∂ρ ′ = 0 . ∂ x j S. Sinayoko, A. Agarwal, Z. Hu Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Goldstein’s theory Non-radiating filter design Equations Sources of sound in an axi-symmetric jet Conclusion Governing equation for fluctuating quantities Conservation of mass for fluctuating quantities ∂ρ ′ ∂ t + ∂ ( ρ v j ) ′ = 0 . (5) ∂ x j Momentum conservation for fluctuating quantities ∂σ ′ + ∂ ( ρ v i v j ) ′ ∂ ( ρ v i ) ′ + ∂ p ′ ij = . (6) ∂ t ∂ x j ∂ x i ∂ x j Taking ∂ ( 6 ) /∂ x i − ∂ ( 5 ) /∂ t gives ∂ 2 σ ′ ∂ t 2 + ∂ 2 ( ρ v i v j ) ′ ∂ 2 p ′ − ∂ 2 ρ ′ ij = . (7) ∂ x i x i ∂ x i ∂ x j ∂ x i ∂ x j S. Sinayoko, A. Agarwal, Z. Hu Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Defining the physical sources of sound Goldstein’s theory Non-radiating filter design Equations Sources of sound in an axi-symmetric jet Conclusion Governing equation for fluctuating quantities ˜ f = ρ f /ρ, Favre averaging, (8) Governing equation j ) = ∂ 2 σ ij ′ ∂ 2 p ′ − ∂ 2 ρ ′ ∂ 2 v j ρ ′ + ρ ˜ v j v ′ v i v ′ (˜ v i ˜ i + ρ ˜ ∂ t 2 + + s (9) ∂ x i ∂ x i ∂ x i ∂ x j ∂ x i ∂ x j Source definition � � ∂ 2 T ij + ρ v ′ i v ′ v i ρ ′ v ′ v j ρ ′ v ′ j + ˜ j + ˜ s = − (10) i ∂ x i ∂ x j T ij = − ρ ( � v i v j − ˜ v i ˜ v j ) . (11) S. Sinayoko, A. Agarwal, Z. Hu Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Problem description Defining the physical sources of sound Filter defining properties Non-radiating filter design Local filter Sources of sound in an axi-symmetric jet Global filter Conclusion Problem description Parallel flow 0 . 8 10 0 . 6 5 y , m 0 0 . 4 − 5 0 . 2 − 10 0 − 50 0 50 100 150 x , m S. Sinayoko, A. Agarwal, Z. Hu Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Problem description Defining the physical sources of sound Filter defining properties Non-radiating filter design Local filter Sources of sound in an axi-symmetric jet Global filter Conclusion Problem description Pressure field 50 y , m 0 − 50 − 50 0 50 100 150 x , m S. Sinayoko, A. Agarwal, Z. Hu Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Problem description Defining the physical sources of sound Filter defining properties Non-radiating filter design Local filter Sources of sound in an axi-symmetric jet Global filter Conclusion Defining properties Fourier transform f ( x , t ) → F ( k , ω ) f ( x , t ) → F ( k , ω ) Non-radiating condition | k | = | ω | F ( k , ω ) = 0 for c ∞ Additional requirement | k | � = | ω | F ( k , ω ) = F ( k , ω ) for c ∞ S. Sinayoko, A. Agarwal, Z. Hu Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Problem description Defining the physical sources of sound Filter defining properties Non-radiating filter design Local filter Sources of sound in an axi-symmetric jet Global filter Conclusion Local filter Filter definition D’Alembertian filter � 1 � ∂ 2 ∂ t 2 − ∇ 2 f ( x , t ) = f ( x , t ) , c 2 ∞ Frequency domain � � | k | 2 − ω 2 F ( k , ω ) = F ( k , ω ) c 2 ∞ | k | = | ω | ⇒ F ( k , ω ) = 0 for c ∞ S. Sinayoko, A. Agarwal, Z. Hu Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Problem description Defining the physical sources of sound Filter defining properties Non-radiating filter design Local filter Sources of sound in an axi-symmetric jet Global filter Conclusion Local filter Results 50 y , m 0 − 50 − 50 0 50 100 150 x , m S. Sinayoko, A. Agarwal, Z. Hu Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Problem description Defining the physical sources of sound Filter defining properties Non-radiating filter design Local filter Sources of sound in an axi-symmetric jet Global filter Conclusion Local filter Results 50 y , m 0 − 50 − 50 0 50 100 150 x , m S. Sinayoko, A. Agarwal, Z. Hu Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Problem description Defining the physical sources of sound Filter defining properties Non-radiating filter design Local filter Sources of sound in an axi-symmetric jet Global filter Conclusion Global filter Filter definition Time-domain Frequency-domain f = w ∗ f (12) F = WF (13) F .T. f ( x , t ) F ( k , ω ) Filter window � 0 if | k | = | ω | / c ∞ 1 otherwise I.F.T. f ( x , t ) F ( k , ω ) S. Sinayoko, A. Agarwal, Z. Hu Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Problem description Defining the physical sources of sound Filter defining properties Non-radiating filter design Local filter Sources of sound in an axi-symmetric jet Global filter Conclusion Global filter Filter definition Gaussian filter � � � � − ( k x − α ) 2 − ( k x + α ) 2 W ( k , ω ) = exp + exp 2 σ 2 2 σ 2 α = 0 . 68 m − 1 , σ = 0 . 1 m − 1 . W ( k , ω ) 4 σ 1 0 k x − α 0 α S. Sinayoko, A. Agarwal, Z. Hu Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Problem description Defining the physical sources of sound Filter defining properties Non-radiating filter design Local filter Sources of sound in an axi-symmetric jet Global filter Conclusion Global filter Results 50 y , m 0 − 50 − 50 0 50 100 150 x , m S. Sinayoko, A. Agarwal, Z. Hu Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Problem description Defining the physical sources of sound Filter defining properties Non-radiating filter design Local filter Sources of sound in an axi-symmetric jet Global filter Conclusion Global filter Results 50 y , m 0 − 50 − 50 0 50 100 150 x , m S. Sinayoko, A. Agarwal, Z. Hu Sep. propagating & non-propagating dynamics in fluid-flow

Introduction Problem description Defining the physical sources of sound Filter defining properties Non-radiating filter design Local filter Sources of sound in an axi-symmetric jet Global filter Conclusion Global filter Validation Comparison with analytical result along profile y = 15m 2 . 5 × 10 − 6 pressure 0 − 2 . 5 × 10 − 6 − 50 0 50 100 150 x , m S. Sinayoko, A. Agarwal, Z. Hu Sep. propagating & non-propagating dynamics in fluid-flow