Development of a Generalized Corcos model for the prediction of turbulent boundary layer induced noise

Transcription

1 Development of a Generalized Corcos model for the prediction of turbulent boundary layer induced noise Anna Caiazzo Roberto D Amico, Wim Desmet KU Leuven - Noise & Vibration Research Group Departement of Mechanical Engineering, Celestijnenlaan 300B, 3001 Heverlee, Belgium. Flinovia II FLow Induced NOise and Vibration Issues and Aspects April 27, State College, PA, USA

2 Outline 1 Introduction 2 Modeling of TBL wall-pressure fluctuations 3 The Generalized Corcos Model 4 Application to a baffled flat plate 5 Conclusions 1 / 20

3 Outline 1 Introduction 2 Modeling of TBL wall-pressure fluctuations 3 The Generalized Corcos Model 4 Application to a baffled flat plate 5 Conclusions 1 / 20

4 Turbulent Boundary Layer (TBL) induced noise: Introduction General Context Automotive industry Aeronautical industry Railway industry Naval industry Turbulent flow noise modelling 2 / 20

5 Turbulent Boundary Layer (TBL) induced noise: Introduction General Context Automotive industry Aeronautical industry Railway industry Naval industry Turbulent flow noise modelling Incompressible RANS + stochastic TBL models Hybrid Methods (Incompressible URANS/ LES/ DES+ Acoustic Analogy) Direct Noise Computation (Compressible LES/DES/URANS/DNS) CPU cost Wall pressure excitation 2 / 20

6 Turbulent Boundary Layer (TBL) induced noise: Introduction General Context Automotive industry Aeronautical industry Railway industry Naval industry Turbulent flow noise modelling Incompressible RANS + stochastic TBL models Hybrid Methods (Incompressible URANS/ LES/ DES+ Acoustic Analogy) Direct Noise Computation (Compressible LES/DES/URANS/DNS) CPU cost EARLY DESIGN STAGE Quick understanding Efficient computation 2 / 20

7 Turbulent Boundary Layer (TBL) induced noise: Introduction General Context Automotive industry Aeronautical industry Railway industry Naval industry Turbulent flow noise modelling Incompressible RANS + stochastic TBL models Hybrid Methods (Incompressible URANS/ LES/ DES+ Acoustic Analogy) Direct Noise Computation (Compressible LES/DES/URANS/DNS) CPU cost EARLY DESIGN STAGE Quick understanding Efficient computation 2 / 20

8 Outline 1 Introduction 2 Modeling of TBL wall-pressure fluctuations Basic assumptions TBL Wavenumber Frequency Spectra 3 The Generalized Corcos Model 4 Application to a baffled flat plate 5 Conclusions 2 / 20

9 Modeling of TBL wall-pressure fluctuations For a flow with U in the x-direction: Basic assumptions 1 Flat rigid surface (x, y) 2 Fully developed TBL 3 Zero mean pressure gradient 4 Low Mach number flow U Pressure fluctuation (t + τ) p x,y Pressure fluctuation p x,y(t) TBL homogeneous in space and stationary in time R pp(ξ x, ξ y, τ) =< p(x, y, t)p(x + ξ x, y + ξ y, t + τ) > 3 / 20

10 Modeling of TBL wall-pressure fluctuations For a flow with U in the x-direction: Basic assumptions 1 Flat rigid surface (x, y) 2 Fully developed TBL 3 Zero mean pressure gradient 4 Low Mach number flow U Pressure fluctuation (t + τ) p x,y Pressure fluctuation p x,y(t) TBL homogeneous in space and stationary in time R pp(ξ x, ξ y, τ) =< p(x, y, t)p(x + ξ x, y + ξ y, t + τ) > 3 / 20

11 Modeling of TBL wall-pressure fluctuations For a flow with U in the x-direction: Basic assumptions 1 Flat rigid surface (x, y) 2 Fully developed TBL 3 Zero mean pressure gradient 4 Low Mach number flow U Pressure fluctuation (t + τ) p x,y Pressure fluctuation p x,y(t) TBL homogeneous in space and stationary in time R pp(ξ x, ξ y, τ) =< p(x, y, t)p(x + ξ x, y + ξ y, t + τ) > 3 / 20

12 Modeling of TBL wall-pressure fluctuations For a flow with U in the x-direction: Basic assumptions 1 Flat rigid surface (x, y) 2 Fully developed TBL 3 Zero mean pressure gradient 4 Low Mach number flow U Pressure fluctuation (t + τ) p x,y Pressure fluctuation p x,y(t) TBL homogeneous in space and stationary in time R pp(ξ x, ξ y, τ) =< p(x, y, t)p(x + ξ x, y + ξ y, t + τ) > Time FT in the physical space ξ: Ψ pp(ξ x, ξ y, ω) Spatial FT in the wavenumber space k: Ψ pp(k x, k y, ω) 3 / 20

13 Modeling of TBL wall-pressure fluctuations For a flow with U in the x-direction: Basic assumptions 1 Flat rigid surface (x, y) 2 Fully developed TBL 3 Zero mean pressure gradient 4 Low Mach number flow U Pressure fluctuation (t + τ) p x,y Pressure fluctuation p x,y(t) TBL homogeneous in space and stationary in time R pp(ξ x, ξ y, τ) =< p(x, y, t)p(x + ξ x, y + ξ y, t + τ) > Use of semi-empirical models of TBL fitted to experimental data 3 / 20

14 Modeling of TBL wall-pressure fluctuations TBL Wavenumber Frequency Spectra Semi-empirical models: Corcos, Chase, Smolyakov-Tkachenko, etc. ( Ψpp(kx,0,ω) ) φ(ω)( Uc ω )2 Normalized Pressure Spectrum [db], 10log Corcos Smol yakovtkachenko Chase 1980 Chase 1987 Range of data from cylinder in water at 6.1 m s 1 Bonness s data (2010) - Spectra level H0 = Range of existing data in air Martin and Leehey Data (1977) - Spectral level H0 = Dimensionless longitudinal wavenumber, Uckx/ω 1 1 N.C. Martin, P. Leehey, Low wavenumber wall pressure measurements using a rectangular membrane as a spatial filter, Journal of Sound and Vibration 52(1)(1977) W. K. Bonness, D.E. Capone, S. A. Hambric, Low wavenumber turbulent boundary layer wall-pressure measurements from vibration data on a cylinder in pipe flow, Journal of Sound and Vibration 329(2010) / 20

15 Modeling of TBL wall-pressure fluctuations TBL Wavenumber Frequency Spectra ( Ψpp(kx,0,ω) ) φ(ω)( Uc ω )2 Normalized Pressure Spectrum [db], 10log Corcos Smol yakovtkachenko Chase 1980 Chase 1987 Range of data from cylinder in water at 6.1 m s 1 Bonness s data (2010) - Spectra level H0 = Range of existing data in air Corcos! Advantageous mathematical features (Space variables separated, convertible model)! Accurate description in the convective( k) domain! Only two free parameters: α x, α y % Small rate of decay at low( k) wavenumbers % Overpredicts the subconvective spectrum ( 20 db higher) Dimensionless longitudinal wavenumber, Uckx/ω Martin and Leehey Data (1977) - Spectral level H0 = / 20

16 Modeling of TBL wall-pressure fluctuations TBL Wavenumber Frequency Spectra ( Ψpp(kx,0,ω) ) φ(ω)( Uc ω )2 Normalized Pressure Spectrum [db], 10log Corcos Smol yakovtkachenko Chase 1980 Chase 1987 Range of data from cylinder in water at 6.1 m s 1 Bonness s data (2010) - Spectra level H0 = Range of existing data in air Corcos! Advantageous mathematical features (Space variables separated, convertible model)! Accurate description in the convective( k) domain! Only two free parameters: α x, α y % Small rate of decay at low( k) wavenumbers % Overpredicts the subconvective spectrum ( 20 db higher) Dimensionless longitudinal wavenumber, Uckx/ω Martin and Leehey Data (1977) - Spectral level H0 = / 20

17 Modeling of TBL wall-pressure fluctuations TBL Wavenumber Frequency Spectra ( Ψpp(kx,0,ω) ) φ(ω)( Uc ω )2 Normalized Pressure Spectrum [db], 10log Corcos Smol yakovtkachenko Chase 1980 Chase 1987 Range of data from cylinder in water at 6.1 m s 1 Bonness s data (2010) - Spectra level H0 = Range of existing data in air Dimensionless longitudinal wavenumber, Uckx/ω Martin and Leehey Data (1977) - Spectral level H0 = Corcos! Advantageous mathematical features (Space variables separated, convertible model)! Accurate description in the convective( k) domain! Only two free parameters: α x, α y % Small rate of decay at low( k) wavenumbers % Overpredicts the subconvective spectrum ( 20 db higher) Chase and Smol yakov Tkachenko! Accurate description in the low-k region - close to experimental data % Lack of simplicity in the mathematical descriptions (Non-separable model) % Many semi-empirical parameters strongly depending on the test conditions. 5 / 20

18 Modeling of TBL wall-pressure fluctuations ( Ψpp(kx,0,ω) ) φ(ω)( Uc ω )2 Normalized Pressure Spectrum [db], 10log Corcos Smol yakovtkachenko Chase 1980 Chase 1987 Bonness s data (2010) - Spectra level H 0 = db Dimensionless longitudinal wavenumber, U ck x/ω Martin and Leehey Data (1977) - Spectral level H 0 = How can we obtain a more accurate low-wavenumber spectrum yet preserving the advantages of the Corcos model? 6 / 20

19 Objectives Development of a new mathematical model built on a two-dimensional Butterworth filter, the Generalized Corcos model, for modelling TBL excitation in vibro-acoustic problems. 2 Using the Generalized Corcos model, it is possible to preserve the main advantages of the known Corcos model, namely a closed form solution with clear benefits on computation time, as well as a good description of the wavenumber-frequency spectrum in the convective domain. In addition, by using a Butterworth-shaped model, it is possible to address the low k limitation found in Corcos and make the spectrum steeper by properly choosing the order of the filter along the two directions. 2 A. Caiazzo, et al.,a Generalized Corcos model for modelling turbulent boundary layer wall pressure fluctuations. Journal of Sound and Vibration 372 (2016) / 20

20 Outline 1 Introduction 2 Modeling of TBL wall-pressure fluctuations 3 The Generalized Corcos Model From the Corcos to the Generalized Corcos model Model comparison 4 Application to a baffled flat plate 5 Conclusions 7 / 20

21 The Generalized Corcos Model From the Corcos to the Generalized Corcos model Step 1: The Corcos wavenumber-frequency spectrum involves two Lorentzian functions to describe the pressure loading: 1 1 πα Ψ pp(k x, k y, ω) = φ(ω) ω πβ ω ( ) 2 ( ) 1 + kx kω ky 2, 1 + α ω β ω k ω = ω/u c, α ω = k ωα x, β ω = k ωα y. { { Lorentzian functions: B 1(k x) B 1(k y) 8 / 20

22 The Generalized Corcos Model From the Corcos to the Generalized Corcos model Step 1: The Corcos wavenumber-frequency spectrum involves two Lorentzian functions to describe the pressure loading: 1 1 πα Ψ pp(k x, k y, ω) = φ(ω) ω πβ ω ( ) 2 ( ) 1 + kx kω ky 2, 1 + α ω β ω k ω = ω/u c, α ω = k ωα x, β ω = k ωα y. { { Lorentzian functions: B 1(k x) B 1(k y) s1,2 = kω ±iαω B1(kx) αω Im(kx/(2π)) kx Re(kx/(2π)) 8 / 20

23 The Generalized Corcos Model From the Corcos to the Generalized Corcos model Step 1: The Corcos wavenumber-frequency spectrum involves two Lorentzian functions to describe the pressure loading: 1 1 πα Ψ pp(k x, k y, ω) = φ(ω) ω πβ ω ( ) 2 ( ) 1 + kx kω ky 2, 1 + α ω β ω k ω = ω/u c, α ω = k ωα x, β ω = k ωα y. { { Lorentzian functions: B 1(k x) B 1(k y) s1,2 = kω ±iαω B1(kx) αω Im(kx/(2π)) kx Re(kx/(2π)) When it comes to the integration over an infinite domain, the Lorentzian function allows mathematical simplifications that keep the Corcos model fully analytical in certain cases! 8 / 20

24 The Generalized Corcos Model From the Corcos to the Generalized Corcos model Step 2: The square magnitude of a Butterworth filter of order one represents a generalization of the Lorentzian function and is characterized by its order: B 1 (k x) = 1 πα ω 1 + ( kx k ω α ω ) 2 B n(k x) = nsin(π/2n) πα ω ) 2n 1 + ( kx k ω α ω (B 1 (k x), B 1 (k y)) (B n(k x), B m(k y)) s k = k ω + α ωe iθ k θ k = π(1 + 2k) 2n 9 / 20

25 The Generalized Corcos Model From the Corcos to the Generalized Corcos model Step 2: The square magnitude of a Butterworth filter of order one represents a generalization of the Lorentzian function and is characterized by its order: B 1 (k x) = 1 πα ω 1 + ( kx k ω α ω ) 2 B n(k x) = nsin(π/2n) πα ω ) 2n 1 + ( kx k ω α ω (B 1 (k x), B 1 (k y)) (B n(k x), B m(k y)) 1 n=1 n=3 n=5 n n = 1 n = 3 n = Bn(kx) αω Im(kx/(2π)) kx Re(kx/(2π)) s k = k ω + α ωe iθ k θ k = π(1 + 2k) 2n 9 / 20

26 The Generalized Corcos Model From the Corcos to the Generalized Corcos model Step 2: The square magnitude of a Butterworth filter represents a generalization of the Lorentzian function and is characterized by its order: Ψ C pp(k x, k y, ω) = φ(ω)b 1 (k x)b 1 (k y) Ψ GC pp (k x, k y, ω) = φ(ω)b n(k x)b m(k y) Spatial IFT + + Ψ pp(ξ x, ξ y, ω) = φ(ω) B n(k x)e ikxξx dk x B m(k y)e ikyξy dk y }{{}}{{} I kx I ky Complex residue theorem Ψ pp(ξ x, ξ y, ω) = φ(ω) Akx A ky π 2 α ωβ ωe ikωξx n 1 nm m 1 e iθ k i ξ y β ωe iθ k k=0 k=0 e iθ k i ξ x α ωe iθ k n = 1, m = 1 Corcos: Ψ pp(ξ x, ξ y, ω) = φ(ω)e ikωξx e ξx αω ξy βω!this allows preserving the analyticity of the integration (separable and convertible model closed form solution) 10 / 20

27 The Generalized Corcos Model From the Corcos to the Generalized Corcos model Step 2: The square magnitude of a Butterworth filter represents a generalization of the Lorentzian function and is characterized by its order: Ψ C pp(k x, k y, ω) = φ(ω)b 1 (k x)b 1 (k y) Ψ GC pp (k x, k y, ω) = φ(ω)b n(k x)b m(k y) Spatial IFT + + Ψ pp(ξ x, ξ y, ω) = φ(ω) B n(k x)e ikxξx dk x B m(k y)e ikyξy dk y }{{}}{{} I kx I ky Complex residue theorem Ψ pp(ξ x, ξ y, ω) = φ(ω) Akx A ky π 2 α ωβ ωe ikωξx n 1 nm m 1 e iθ k i ξ y β ωe iθ k k=0 k=0 e iθ k i ξ x α ωe iθ k n = 1, m = 1 Corcos: Ψ pp(ξ x, ξ y, ω) = φ(ω)e ikωξx e ξx αω ξy βω!this allows preserving the analyticity of the integration (separable and convertible model closed form solution) 10 / 20

28 The Generalized Corcos Model From the Corcos to the Generalized Corcos model Step 2: The square magnitude of a Butterworth filter represents a generalization of the Lorentzian function and is characterized by its order: Ψ C pp(k x, k y, ω) = φ(ω)b 1 (k x)b 1 (k y) Ψ GC pp (k x, k y, ω) = φ(ω)b n(k x)b m(k y) Spatial IFT + + Ψ pp(ξ x, ξ y, ω) = φ(ω) B n(k x)e ikxξx dk x B m(k y)e ikyξy dk y }{{}}{{} I kx I ky Complex residue theorem Ψ pp(ξ x, ξ y, ω) = φ(ω) Akx A ky π 2 α ωβ ωe ikωξx n 1 nm m 1 e iθ k i ξ y β ωe iθ k k=0 k=0 e iθ k i ξ x α ωe iθ k n = 1, m = 1 Corcos: Ψ pp(ξ x, ξ y, ω) = φ(ω)e ikωξx e ξx αω ξy βω!this allows preserving the analyticity of the integration (separable and convertible model closed form solution) 10 / 20

29 The Generalized Corcos Model From the Corcos to the Generalized Corcos model Step 2: The square magnitude of a Butterworth filter represents a generalization of the Lorentzian function and is characterized by its order: Ψ C pp(k x, k y, ω) = φ(ω)b 1 (k x)b 1 (k y) Ψ GC pp (k x, k y, ω) = φ(ω)b n(k x)b m(k y) Spatial IFT + + Ψ pp(ξ x, ξ y, ω) = φ(ω) B n(k x)e ikxξx dk x B m(k y)e ikyξy dk y }{{}}{{} I kx I ky Complex residue theorem Ψ pp(ξ x, ξ y, ω) = φ(ω) Akx A ky π 2 α ωβ ωe ikωξx n 1 nm m 1 e iθ k i ξ y β ωe iθ k k=0 k=0 e iθ k i ξ x α ωe iθ k n = 1, m = 1 Corcos: Ψ pp(ξ x, ξ y, ω) = φ(ω)e ikωξx e ξx αω ξy βω!this allows preserving the analyticity of the integration (separable and convertible model closed form solution) 10 / 20

30 The Generalized Corcos Model From the Corcos to the Generalized Corcos model Step 2: The square magnitude of a Butterworth filter represents a generalization of the Lorentzian function and is characterized by its order: Ψ C pp(k x, k y, ω) = φ(ω)b 1 (k x)b 1 (k y) Ψ GC pp (k x, k y, ω) = φ(ω)b n(k x)b m(k y) Spatial IFT + + Ψ pp(ξ x, ξ y, ω) = φ(ω) B n(k x)e ikxξx dk x B m(k y)e ikyξy dk y }{{}}{{} I kx I ky Complex residue theorem Ψ pp(ξ x, ξ y, ω) = φ(ω) Akx A ky π 2 α ωβ ωe ikωξx n 1 nm m 1 e iθ k i ξ y β ωe iθ k k=0 k=0 e iθ k i ξ x α ωe iθ k n = 1, m = 1 Corcos: Ψ pp(ξ x, ξ y, ω) = φ(ω)e ikωξx e ξx αω ξy βω!this allows preserving the analyticity of the integration (separable and convertible model closed form solution) 10 / 20

31 The Generalized Corcos Model From the Corcos to the Generalized Corcos model Step 2: The square magnitude of a Butterworth filter represents a generalization of the Lorentzian function and is characterized by its order: Ψ C pp(k x, k y, ω) = φ(ω)b 1 (k x)b 1 (k y) Ψ GC pp (k x, k y, ω) = φ(ω)b n(k x)b m(k y) Spatial IFT + + Ψ pp(ξ x, ξ y, ω) = φ(ω) B n(k x)e ikxξx dk x B m(k y)e ikyξy dk y }{{}}{{} I kx I ky Complex residue theorem Ψ pp(ξ x, ξ y, ω) = φ(ω) Akx A ky π 2 α ωβ ωe ikωξx n 1 nm m 1 e iθ k i ξ y β ωe iθ k k=0 k=0 e iθ k i ξ x α ωe iθ k n = 1, m = 1 Corcos: Ψ pp(ξ x, ξ y, ω) = φ(ω)e ikωξx e ξx αω ξy βω!this allows preserving the analyticity of the integration (separable and convertible model closed form solution) 10 / 20

32 The Generalized Corcos Model TBL Wavenumber Frequency Spectra + By properly choosing the order of the filter, n and m, it is possible to control the decay of the TBL model. Φpp(kx,0,ω) Normalized Wavevector Spectrum in k x for U = 35.8 m s 1,f = 500 Hz m n fixed n m fixed 4B n(k x)b m(k y) Φ pp(k x, k y, ω) = n 1 k=0 e iθ k m 1 k=0 e iθ k ω 2 /U 2 c ( ) Akx A ky αω βω n = 1, m = 1 Corcos: 2α x Φ pp(k x, k y, ω) = [ ( ) ] α 2 x + Uckx 2 ω 1 nm Dimensionless longitudinal wavenumber, U ck x/ω 2α y [ ( α 2 y + Ucky ) ] 2 ω Corcos ( ) ; Generalized Corcos with m = 1 n = 1 ( ), m = 1 n = 2 ( ), m = 2 n = 1 ( ), m = 2 n = 2 ( ). 11 / 20

33 The Generalized Corcos Model Model comparison ( Ψpp(kx,0,ω) ) φ(ω)( Uc ω )2 Normalized Pressure Spectrum [db], 10log Corcos Generalized Corcos Model m=1 n=2 Smol yakovtkachenko Chase 1980 Chase 1987 incompressible Dimensionless longitudinal wavenumber, Uckx/ω Generalized Corcos ky kx (α x, α y) = (0.115, 0.7) U = 35.8 m s 1 [db] !Improve accuracy at low wavenumber over the original Corcos model!allows closed form expression for the response of simple structures (i.e. simply supported flat plates)!control the decay of the TBL model by changing n and m!steeper curve increasing n, along the streamwise direction 12 / 20

34 Outline 1 Introduction 2 Modeling of TBL wall-pressure fluctuations 3 The Generalized Corcos Model 4 Application to a baffled flat plate Model problem & Results Effect of different orders of the filter Computation time 5 Conclusions 12 / 20

35 Application to a baffled flat plate Model problem 1 Structural model: Simply supported plate 2 Aerodynamic model: TBL fully developed, stationary and homogeneous Cross-spectrum: Corcos, Generalized Corcos with (α x, α y) = (0.10, 0.77), Chase 1980 and Smol yakov-tkachenko y Turbulent boundary layer z a (ρ 1, c 1) x (ρ 0, c 0) b Infinite baffle 3 P(x s; t) = p ac(x s; t) p TBL (x s; t) 6 Test cases: 4 Weak coupling approximation 5 Modal expansion method: exact spectral response. 13 / 20

36 Application to a baffled flat plate Model problem 1 Structural model: Simply supported plate 2 Aerodynamic model: TBL fully developed, stationary and homogeneous Cross-spectrum: Corcos, Generalized Corcos with (α x, α y) = (0.10, 0.77), Chase 1980 and Smol yakov-tkachenko y Turbulent boundary layer z a (ρ 1, c 1) x (ρ 0, c 0) b Infinite baffle 3 P(x s; t) = p ac(x s; t) p TBL (x s; t) 6 Test cases: 4 Weak coupling approximation 5 Modal expansion method: exact spectral response. 13 / 20

37 Application to a baffled flat plate Model problem 1 Structural model: Simply supported plate 2 Aerodynamic model: TBL fully developed, stationary and homogeneous Cross-spectrum: Corcos, Generalized Corcos with (α x, α y) = (0.10, 0.77), Chase 1980 and Smol yakov-tkachenko y Turbulent boundary layer z a (ρ 1, c 1) x (ρ 0, c 0) b Infinite baffle 3 P(x s; t) = p ac(x s; t) p TBL (x s; t) 6 Test cases: 4 Weak coupling approximation 5 Modal expansion method: exact spectral response. 13 / 20

38 Application to a baffled flat plate Model problem 1 Structural model: Simply supported plate 2 Aerodynamic model: TBL fully developed, stationary and homogeneous Cross-spectrum: Corcos, Generalized Corcos with (α x, α y) = (0.10, 0.77), Chase 1980 and Smol yakov-tkachenko y Turbulent boundary layer z a (ρ 1, c 1) x (ρ 0, c 0) b Infinite baffle 3 P(x s; t) = p ac(x s; t) p TBL (x s; t) 6 Test cases: 4 Weak coupling approximation 5 Modal expansion method: exact spectral response. 13 / 20

39 Application to a baffled flat plate Model problem 1 Structural model: Simply supported plate 2 Aerodynamic model: TBL fully developed, stationary and homogeneous Cross-spectrum: Corcos, Generalized Corcos with (α x, α y) = (0.10, 0.77), Chase 1980 and Smol yakov-tkachenko y Turbulent boundary layer z a (ρ 1, c 1) 6 Test cases: x (ρ 0, c 0) b Infinite baffle 3 P(x s; t) = p ac(x s; t) p TBL (x s; t) 4 Weak coupling approximation (a) U = 140 m s 1 5 Modal expansion method: exact spectral response. 13 / 20

40 Application to a baffled flat plate Model problem 1 Structural model: Simply supported plate 2 Aerodynamic model: TBL fully developed, stationary and homogeneous Cross-spectrum: Corcos, Generalized Corcos with (α x, α y) = (0.10, 0.77), Chase 1980 and Smol yakov-tkachenko y Turbulent boundary layer z a (ρ 1, c 1) 6 Test cases: x (ρ 0, c 0) b Infinite baffle 3 P(x s; t) = p ac(x s; t) p TBL (x s; t) (a) U = 140 m s 1 4 Weak coupling approximation (b) U = 240 m s 1 5 Modal expansion method: exact spectral response. 13 / 20

41 1 Structural model: Application to a baffled flat plate Model problem Simply supported plate 2 Aerodynamic model: TBL fully developed, stationary and homogeneous Cross-spectrum: Corcos, Generalized Corcos with (α x, α y) = (0.10, 0.77), Chase 1980 and Smol yakov-tkachenko y Turbulent boundary layer z a (ρ 1, c 1) 6 Test cases: x (ρ 0, c 0) b Infinite baffle 3 P(x s; t) = p ac(x s; t) p TBL (x s; t) (a) U = 140 m s 1 4 Weak coupling approximation (b) U = 240 m s 1 5 Modal expansion method: exact spectral response. Sound power radiated PSD S 1 (ω) = 2ξ f 1 r,s ( ) Re Zrs 1 Φrs d rs 2 13 / 20

42 Application to a baffled flat plate Results - Case (a) U = 140 m s 1 (λ ω < λ s), modes weakly driven by the boundary layer Sub-convective region of Φ pp(k; ω) Suitable TBL model: Chase and Smol yakov-t Low k Φpp(kx,0,ω) Dimensionless longitudinal wavenumber, Uckx/ω Normalized Wavevector Spectrum for U = 140 m s 1 at 1 khz. 14 / 20

43 U = 140 m s 1 Application to a baffled flat plate Results - Case (a) (λ ω < λ s), modes weakly driven by the boundary layer Sub-convective region of Φ pp(k; ω) Suitable TBL model: Chase and Smol yakov-t Φpp(kx,0,ω) 10 2 Low k Dimensionless longitudinal wavenumber, Uckx/ω Normalized Wavevector Spectrum for U = 140 m s 1 at 1 khz. Radiated Sound Power Spectrum, S1(ω) U = 140 m s Frequency [Hz] Corcos Smol yakov Tkachenko Generalized Corcos Model m=1 n=2 Chase 1980!The low-k spectra seen for the Generalized Corcos model with (n, m) = (2, 1) lie at a lower radiated sound power than Corcos, getting close to Chase: improved accuracy at low-k. 14 / 20

44 Application to a baffled flat plate Results - Case (b) U = 240 m s 1 modes strongly driven by the boundary layer (hydrodynamic coincidence) Convective region of Φ pp(k; ω) Suitable TBL model: Corcos Φpp(kx,0,ω) Dimensionless longitudinal wavenumber, Uckx/ω Normalized Wavevector Spectrum for U = 240 m s 1 at 2 khz. 15 / 20

45 U = 240 m s 1 Application to a baffled flat plate modes strongly driven by the boundary layer (hydrodynamic coincidence) Convective region of Φ pp(k; ω) Suitable TBL model: Corcos Results - Case (b) Φpp(kx,0,ω) Dimensionless longitudinal wavenumber, Uckx/ω Normalized Wavevector Spectrum for U = 240 m s 1 at 2 khz. Radiated Sound Power Spectrum, S1(ω) U = 240 m s Frequency [Hz] Corcos Chase 1980 Smol yakov-tkachenko Generalized Corcos Model m=1 n=2!by using the Generalized Corcos model with (n, m) = (2, 1) the excitation levels, determined by the convective peak, are still close to the one given by Corcos. 15 / 20

46 Effect of different orders of the filter Results - Case (a) CORCOS U = 140 m s Φpp(kx,0,ω) S1(ω) CHASE Dimensionless longitudinal wavenumber, Uckx/ω Normalized Wavevector Spectrum for U = 140 m s 1 at 1 khz Frequency [Hz] Radiated sound power spectra Corcos ( ); Chase ( ); Generalized Corcos model m = 1 n = 1 ( ), m = 1 n = 2 ( ), m = 2 n = 1 (- -), m = 2 n = 2 ( ). (1) By properly choosing the order of the filter,(n, m), it is possible to reduce the Corcos spectrum in the entire wavenumber range outside the convective ridge. Either reaching the excitation level given by Corcos, (n, m) = (1, 1), or lower getting close to Chase and Smol yakov-t. 16 / 20

47 The Generalized Corcos model is found to be close to that of Chase in the low k range. Best fit for n = 2 and m = 1, 2 Effect of different orders of the filter Φpp(kx,0,ω) Best fit for the Chase model - Case (a) Chase 1980 GC m=1 n=2 GC m=2 n=3 GC m=2 n=2 freq= 1 khz 10 3 Radiated Sound Power Spectrum, S1(ω) Frequency [Hz] Dimensionless longitudinal wavenumber, Uckx/ω U = 140 m s 1 Chase 1980 GC m=2 n=3 GC m=2 n=2 GC m=1 n=2 CHASE 17 / 20

48 Effect of different orders of the filter Best fit for the Smol yakov-tkachenko model - Case (a) The Generalized Corcos model is found to be close to that of Smol yakov-tkachenko in the low k range. Best fit for n = 3 and m = 1, 2, 3 Φpp(kx,0,ω) Smol yakovtkachenko GC n=2 m=1 GC n=3 m=1 GC n=3 m=2 GC n=3 m=3 freq= 1 khz Dimensionless longitudinal wavenumber, Uckx/ω Radiated Sound Power Spectrum, S1(ω) U = 140 m s 1 GC n=3 m=1 GC n=3 m=2 Smol yakov Tkachenko 10 3 Frequency [Hz] GC n=3 m=3 GC n=2 m=1 Smol yakov Tkachenko 18 / 20

49 Computation time Sound power radiated PSD S 1 (ω) = 2ξ f 1 r,s ( ) Re Zrs 1 Φrs d rs 2 Modal excitation term Φ rs(ω) = 1 (2π) Φ pp(k x, k y, ω) S rs(k x, k y) 2 dk x dk y Chase, Smol yakov-tkachenko Numerical integration for Φ rs(ω): CPU Corcos, Generalized Corcos Closed form expression for Φ rs(ω): CPU 10 4 s Modal excitation term for the Generalized Corcos model ( ω 2 ) Φ r Φs Φ rs(ω) = U 2 c m 1 k=0 e iθ k n 1 k=0 e iθ k (2) Closed form expression for Φ rs(ω): reduced computational effort to employ CPU 19 / 20

50 Outline 1 Introduction 2 Modeling of TBL wall-pressure fluctuations 3 The Generalized Corcos Model 4 Application to a baffled flat plate 5 Conclusions 19 / 20

51 Conclusions Analytical development of a new TBL model, called the Generalized Corcos model, based on a Butterworth filter formulation in two dimensions 3 Analysis of the position of the new model with respect to the existing TBL models by looking to different application cases and its possible applications. Advantages of the Generalized Corcos model! Separable and convertible model closed form solution the computational effort is definitively reduced! Two additional parameters, n and m, to control the decay of the TBL model, and make it steeper if necessary.! Approach of practical interest as it provides an accurate description of the wavenumber-frequency spectrum at and below the convective peak 3 A. Caiazzo, et al.,a Generalized Corcos model for modelling turbulent boundary layer wall pressure fluctuations. Journal of Sound and Vibration 372 (2016) / 20

52 Conclusions Analytical development of a new TBL model, called the Generalized Corcos model, based on a Butterworth filter formulation in two dimensions 3 Analysis of the position of the new model with respect to the existing TBL models by looking to different application cases and its possible applications. Advantages of the Generalized Corcos model! Separable and convertible model closed form solution the computational effort is definitively reduced! Two additional parameters, n and m, to control the decay of the TBL model, and make it steeper if necessary.! Approach of practical interest as it provides an accurate description of the wavenumber-frequency spectrum at and below the convective peak Experimental validation Open Issues Extending its application to more complex cases 3 A. Caiazzo, et al.,a Generalized Corcos model for modelling turbulent boundary layer wall pressure fluctuations. Journal of Sound and Vibration 372 (2016) / 20

53 Thank you for your attention! Thank You Flinovia II FLow Induced NOise and Vibration Issues and Aspects Contact information: Anna Caiazzo 20 / 20