Computational aeroacoustics

Transcription

1 Turbulence and Aeroacoustics Research team of the Centre Acoustique École Centrale de Lyon & LMFA UMR CNRS 5509 Airbus - Toulouse, March 19-20, 2013 Computational aeroacoustics Christophe Bailly, Christophe Bogey & Olivier Marsden Université de Lyon Ecole Centrale de Lyon & LMFA - UMR CNRS 5509 & Institut Universitaire de France 1

2 Outline Day 1 (4 x 1h30) General introduction and goals of this seminar ( 1/2h) Simulation of turbulent flows ( 2h30) Numerical methods ( 3h) Day 2 (2 x 1h30) From CFD to CAA Formulations informed by steady / unsteady CFD ( 2h30) Strategies in CAA ( 1/2h) 2 Airbus - EEA Faculty - Computational Aeroacoustics

3 Turbulent flows and aeroacoustics in a nutshell! 3 Airbus - EEA Faculty - Computational Aeroacoustics

4 x Turbulent flows q 4 Airbus - EEA Faculty - Computational Aeroacoustics

5 Turbulent flows Free shear flows : subsonic turbulent jets Reynolds number Re D = u j D/ν Prasad & Sreenivasan (1989) Re D 4000 Dimotakis et al. (1983) Re D 10 4 Kurima, Kasagi & Hirata (1983) Ayrault, Balint & Schon (1981) Mollo-Christensen (1963) Re D Re D Re D = Airbus - EEA Faculty - Computational Aeroacoustics

6 Numerical simulation of turbulent flows Turbulence theory large scales L, u Re L = u L ν = L2 /ν L/u viscous time convective time u L Kolmogorov scales (smallest scales) l η, u η Re lη = u ηl η ν 1 Tavoularis (2003) passive scalar mixing Sc 2000 energy input to create turbulence (power per unit volume) de c dt ρu 2 ρu 3 L/u L dissipation of energy (power per unit volume) at the smallest scales ρε µ u2 η l 2 η 6 Airbus - EEA Faculty - Computational Aeroacoustics

7 Numerical simulation of turbulent flows Turbulence theory Balance between production and dissipation ρu 3 L µu2 η l 2 η = L/l η Re 3/4 L Direct Numerical Simulation (DNS), x l η Total number of grid points in 1-D, n x Re 3/4 L and in 3-D, n dof n 3 x Re 9/4 L Arnold Kolmogorov ( ) Time step t x/c Total cost n t T c t n dof n t Re 3 L L u t Lc u x Re3/4 L 1 M a 1 M a Re 3 L (incompressible) 7 Airbus - EEA Faculty - Computational Aeroacoustics

8 Numerical simulation of turbulent flows Yokokawa, Itukara, Ishihara & Kaneda (2002), Kaneda & Ishihara (2006) Isotropic turbulence, & , Re λg = 1217 (Earth Simulator, Japon) ( )l η ( )l η 8 Airbus - EEA Faculty - Computational Aeroacoustics

9 Numerical simulation of turbulent flows Yokokawa, Itukara, Ishihara & Kaneda (2002) ( )l η ( )l η 9 Airbus - EEA Faculty - Computational Aeroacoustics

10 Dynamics of isotropic turbulence Theory of Kolmogorov : energy cascade 2 E11(k1)/(ǫν 5 ) 1/ k 5/3 1 Pao (1965) k 1 /k η Re λg u 2 1 = + E (1) 11 (k 1)dk 1 23 boundary layer (Tielman, 1967) 23 cylinder wake (Uberoi & Freymuth, 1969) 37 grid turbulence (Comte Bellot & Corrsin, 1971) 72 grid turbulence (Comte Bellot & Corrsin, 1971) 130 homogeneous shear flow (Champagne et al., 1970) pipe flow (Laufer, 1952) 282 boundary layer (Tielman, 1969) 308 cylinder wake (Uberoi & Freymuth, 1969) 401 boundary layer (Sanborn & Marshall, 1965) 540 grid turbulence (Kistler & Vrebalovich, 1966) 600 boundary layer (Saddoughi, 1994) 780 round jet (Gibson, 1963) 850 boundary layer (Coantic & Favre, 1974) 1500 boundary layer (Saddoughi, 1994) 2000 tidal channel (Grant et al., 1962) 3180 return channel (CAHI Moscou, 1991) 10 Airbus - EEA Faculty - Computational Aeroacoustics

11 Dynamics of isotropic turbulence Theory of Kolmogorov : energy cascade E (k) = ε = u 3 L f imposed E (k) C Kε 2/3 k 5/3 k 2 E (k) k t = ε = 2ν 0 0 E(k)dk k 2 E(k)dk O dissipation 1 L f 1 λ g 1 l η k L f /l η Re 3/4 L f l η = ν 3/4 ε 1/4 11 Airbus - EEA Faculty - Computational Aeroacoustics

12 Dynamics of isotropic turbulence Theory of Kolmogorov : energy cascade small k intermediate k high k 12 Airbus - EEA Faculty - Computational Aeroacoustics

13 Statistical tools & signal processing Subsonic round jet flow nozzle diameter D = 0.08 m, exit velocity U j = 19.8 m.s 1 (data Olivier Marsden) Reynolds number Re D 10 5 u 1 (t) at x/d u 1 (m/s) 0 pdf t (s) u 1 (m/s) u m.s 1 u 1 /U j 0.07 u 1 /Ū S u T u 1 = Airbus - EEA Faculty - Computational Aeroacoustics

14 Statistical tools & signal processing Subsonic round jet flow nozzle diameter D = 0.08 m, exit velocity U j = 19.8 m.s 1 (data Olivier Marsden) Reynolds number Re D 10 5 u 1 (t) at x/d u 1 (m/s) 0 pdf t (s) u 1 (m/s) u m.s 1 u 1 /U j 0.13 u 1 /Ū S u T u 1 = Airbus - EEA Faculty - Computational Aeroacoustics

15 Statistical tools & signal processing Subsonic round jet flow (t) at x/d 11.9 u R(τ) φ(f) τ(s) f(hz) f s = 1/ t = 25 khz, n fft = 4096, f = f s /n fft = 6.1 Hz Θ s, L f Ū 1 Θ m, l η m, f η 9.1 khz 15 Airbus - EEA Faculty - Computational Aeroacoustics

16 Direct Numerical Simulation of a subsonic round jet M = 0.9 & Re D = Numerical simulation of turbulent flows 4 NEC SX-8 cluster at HLRS center in Stuttgart, Germany 212 GFlops, 250 GB of memory, 30,000 CPU hours 16 vorticity norm in the plane z = 0 full jet in top view 0 x 27.5r 0 (116 pts bottom figure) n x n y n z = pts k c η 1.5 k c grid cut-off wavenumber x = y = z = r 0 /68 simulation time T = 3000r 0 /u j, 295,000 iterations δ θ = 0.01 r 0 Bogey & Marsden in Advances in Parallel Computing, 19, 2009 see also Bogey & Bailly, J. Fluid Mech., 629, Airbus - EEA Faculty - Computational Aeroacoustics

17 Turbulence Textbooks Batchelor, G.K., 1967, An introduction to fluid dynamics, Cambridge University Press, Cambridge. Bailly, C. & Comte Bellot, G., 2003 Turbulence, CNRS éditions, Paris. Candel, S., 1995, Mécanique des fluides, Dunod Université, 2nd édition, Paris. Davidson, P. A., 2004, Turbulence. An introduction for scientists and engineers, Oxford University Press, Oxford. Davidson, P.A., Kaneda, Y., Moffatt, H.K. & Sreenivasan, K.R., Edts, 2011, A voyage through Turbulence, Cambridge University Press, Cambridge. Guyon, E., Hulin, J.P. & Petit, L., 2001, Hydrodynamique physique, EDP Sciences / Editions du CNRS, première édition 1991, Paris - Meudon (translated in english). Hinze, J.O., 1975, Turbulence, McGraw-Hill International Book Company, New York, 1 ère édition en Landau, L. & Lifchitz, E., 1971, Mécanique des fluides, Editions MIR, Moscou. Also Pergamon Press, 2nd edition, Lesieur, M., 2008, Turbulence in fluids : stochastic and numerical modelling, Kluwer Academic Publishers, 4th revised and enlarged ed., Springer. Pope, S.B., 2000, Turbulent flows, Cambridge University Press. Tennekes, H. & Lumley, J.L., 1972, A first course in turbulence, MIT Press, Cambridge, Massachussetts. Van Dyke, M., 1982, An album of fluid motion, The Parabolic Press, Stanford, California. White, F., 1991, Viscous flow, McGraw-Hill, Inc., New-York, first edition Airbus - EEA Faculty - Computational Aeroacoustics

18 Computational Aeroacoustics Characteristic scales sound field λ a Strouhal number St = fl s U Mach number M a = U or U j c turbulent flow l s Ratio between the source size l s and the acoustic wavelength λ a, λ a l s = c U U fl s = 1 M a 1 St As M a 0, λ a /l s and sources are compact. In any case however, l s remains larger than the characteristic length scale of the shear flow which produces turbulence, e.g. momentum thickness δ θ for a boundary layer or a jet. 18 Airbus - EEA Faculty - Computational Aeroacoustics

19 Computational Aeroacoustics Low-Reynolds number flow around a cylinder Instantaneous fluctuating pressure field Re D = 150 and M a 0.33 λ a 16.5D St = f(re D ) St D U λ a D = 1 M a St D D = m C p = p w p ρ U 2 / Marsden et al., J. Comput. Acoust., 13(4), Airbus - EEA Faculty - Computational Aeroacoustics

20 Computational Aeroacoustics Laminar flow around a 2-D NACA 0012 airfoil M a = 0.29, Re c = 33300, chord c = m, pressure and vorticity fields λ a c vorticity ω z, levels ± s 1 St = fc/u 0.38 St = f 2δ/U 0.24 δ 4.92c/Re 1/2 c (Roshko, 1955 ; Paterson et al., 1973) 20 Airbus - EEA Faculty - Computational Aeroacoustics

21 Computational Aeroacoustics Disparity of scales Isothermal jet, λ a /δ θ Re D /(M a St) (Re D = 10 6, M a = 0.9, r/d 10) p 2 a θ=90 o M7.5 a u a/u 10 4 p a/p 10 3 u a p a λ a δ θ u j D u p x c λ a /δ θ 10 3 u /u j 0.16 Mollo-Christensen (1963), Re D = laminar potential core length x c 21 Airbus - EEA Faculty - Computational Aeroacoustics

22 Computational Aeroacoustics Turbulence and sound generation (aerodynamic noise) Numerical constraints Direct computation of aerodynamic noise (DNC) x l η Total number of grid points in 1-D, n x L + λ ) a Re 3/4 L (1 + 1Ma l η λ a L = u j c = fl u j 1 St M a 1 M a Time step t x/c n t T c t L u t Lc u x Re3/4 L 1 M a Total cost n dof n t Re 3 L 1 M a ( M a ) 3 Re 3 L 1 M 4 a, M a 1 22 Airbus - EEA Faculty - Computational Aeroacoustics

23 Aeroacoustic scaling Von Braun ( ) / Saturn V Acoustic Mach number M a M a = u j c noise M n a Reynolds number Re D Re D = u jd ν = D2 /ν D/u j viscous time convective time D u j nozzle Strouhal number St St = fd u j = f u j /D non-dimension frequency 23 Airbus - EEA Faculty - Computational Aeroacoustics

24 CFD versus CAA Apparently different disciplines CFD Computational Fluid Dynamics CAA Computational AeroAcoustics, related to propagation/radiation of sound DNC Direct Noise Computation, when CAA CFD! LES DNS CFD Z/D DES RANS 24 Airbus - EEA Faculty - Computational Aeroacoustics

25 Computational Aeroacoustics Different levels of representation/modelling in aeroacoustics analogies hybrid methods unsteady DNC & WEM resolved physics & computational cost incompressible CFD compressible statistical models steady DNC = Direct Noise Computation WEM = Wave Extrapolation Methods 25 Airbus - EEA Faculty - Computational Aeroacoustics

26 Numerical simulation of turbulent flows high-performance computing 26 Airbus - EEA Faculty - Computational Aeroacoustics

27 Numerical simulation of turbulent flows Three / four main classes of methods Direct Numerical Simulation (DNS) The whole spectrum up to the Kolmogorov (dissipative) scale is resolved, reference simulations. Statistical approaches (RANS) The Reynolds Averaged Navier-Stokes (RANS) equations are solved : need of a turbulence model, e.g. k t ε, k t ω SST, R ij ε or SA-ν t Large Eddy Simulations (LES) and hybrid approaches (DES, DDES or ZDES) The filtered Navier-Stokes equations are solved : need of a subgrid-scale model e.g. (dynamic) Smagorinsky, ADM, RF-LES,... Lattice Boltzmann method (LBM) Mesoscopic kinetic equation for particle distribution is solved, and primitive variables are obtained from these distribution functions 27 Airbus - EEA Faculty - Computational Aeroacoustics

28 Statistical description of turbulent flows Computed mean flow field in a diaphragm (standard k t ε model) Duct height 2.29h & width w = 2.86h, h = 35 mm obstruction height, Re h = Ūmh/ν = Streamlines in xy-planes computed with the Ū1 and Ū2 components of the 3-D mean velocity field at three spanwise locations z/w = 0.1, 0.5 and 0.9. Mélanie Piellard (Ph.D. Thesis, ECL - DELPHI, 2008) 28 Airbus - EEA Faculty - Computational Aeroacoustics

29 Statistical description of turbulent flows The statistical mean F(x, t) of a variable f(x, t) is then defined as F(x, t) = lim N 1 N N f (i) (x, t) where f (i) is the i-th realization : convenient when manipulating equations but difficult to be implemented in practice! i=1 Temporal average F(x) = lim T 1 T t0 +T t 0 f(x, t )dt Spatial average F(t) = lim V 1 V V f(x, t)dx for a stationary turbulent field, i.e. when time t does not enter into F (hypothesis of ergodicity) for a homogeneous turbulent field, i.e. when position x does not enter into F The statistical mean is a linear operator, which commutes with time and space derivative operators : rules of Reynolds. 29 Airbus - EEA Faculty - Computational Aeroacoustics

30 Statistical description of turbulent flows Some important properties f F + f with f = 0 (and thus f = f F, f = F F = 0) Product of two variables f and g, fg ( F + f )(Ḡ + g ) = FḠ + Fg + f Ḡ + f g and thus, fg = F Ḡ + F g + f Ḡ + f g = F Ḡ + f g f g is a new second-moment term The Reynolds decomposition u i Ū i + u i with u i = 0 Ū i u i part which can be reasonably calculated part which must be modelled (random fluctuations) Assumptions (to simplify) : incompressible flow, u = 0, and homogeneous fluid, constant density ρ. 30 Airbus - EEA Faculty - Computational Aeroacoustics

31 Statistical description of turbulent flows The Reynolds averaged Navier-Stokes equations Navier-Stokes equations (ρu i ) t u i x i = 0 + x j ( ρui u j ) = p x i + τ ij x j τ ij = 2µs ij The Reynolds decomposition u i Ū i + u i p P + p τ ij τ ij + τ ij Ū i x i = 0 ( ) u i = 0 x i (ρū i ) t + (ρū i Ū j ) x j = P + ( ) τ ij ρu i x i x u j j 31 Airbus - EEA Faculty - Computational Aeroacoustics

32 Statistical description of turbulent flows Reynolds Averaged Navier-Stokes (RANS) equations ρu i u j Reynolds stress tensor (new unknown) Generally this term is larger than the mean viscous stress tensor except for wall bounded flows, where viscosity effects become preponderant close to the wall. Total strain seen by the fluid, τ t = τ ij ρu i u j closure problem for ρu i u j by writting a new equation for ρu i u j by modelling the Reynolds tensor Boussinesq s hypothesis, introduction of a turbulent viscosity µ t (ρū i ) t ρu i u j = 2µ t S ij 2 3 ρk tδ ij + (ρū i Ū j ) x j = ( P + 2 ) x i 3 ρk t k t u iu i 2 + x j [ 2(µ + µ t ) S ij ] 32 Airbus - EEA Faculty - Computational Aeroacoustics

33 Numerical simulation of turbulent flows The k t ε turbulence model Almost all the physics is contained in ν t. [ Dimensionally, ν t length velocity m 2.s 1] Two scales are required to evaluate ν t (or transport equation on ν t, SA) k t ε turbulence model length k 3/2 t /ε velocity k 1/2 t ν t C µ k 2 t ε (C µ = cte) Need of the transport equations of k t (okay!) and ε (tricky problem) Usually, high-reynolds number assumption ρε = τ ij u i x j ρε h = µ u i x j u i x j as Re (ε h dissipation rate for an homogeneous turbulence) 33 Airbus - EEA Faculty - Computational Aeroacoustics

34 Numerical simulation of turbulent flows Standard k t ε turbulence model (high-reynolds number form) Jones & Launder (1972), Launder & Spalding (1974) d dt (ρk t) = x j d dt (ρε) = x j [( µ + µ t σ k [( µ + µ t σ ε ) ] kt + P ρε x j ) ] ε + ε (C ε1 P C ε2 ρε) x j k t d F dt F t + x i ( FŪ i ) P = ρu i u j Ū i x j µ t = C µ ρ k2 t ε Constants are determined from canonical experiments C µ = 0.09, C ε1 = 1.44, C ε2 = 1.92, σ k = 1.0, σ ε = 1.3 Refer to lecture notes : low-reynolds number form, Favre average & compressible form 34 Airbus - EEA Faculty - Computational Aeroacoustics

35 Numerical simulation of turbulent flows k t ε turbulence model with compressible corrections Supersonic jet M j = 1.33, T j /T = 1 (estet - EDF) Bailly, Lafon & Candel, J. Sound Vib. (1996) 35 Airbus - EEA Faculty - Computational Aeroacoustics

36 Numerical simulation of turbulent flows k t ε turbulence model with compressible corrections Supersonic jet M j = 2.0, T j /T = 1 (estet - EDF) Bailly, Lafon & Candel, J. Sound Vib. (1996) 36 Airbus - EEA Faculty - Computational Aeroacoustics

37 Numerical simulation of turbulent flows Realizability conditions Galilean invariance & thermodynamics Realizability conditions for the Reynolds stress tensor (no summation for Greek indices) u αu α 0 u αu β2 u α u α u β u β det(u αu β ) 0 De Vachat, Phys. Fluids (1977) Schumann, Phys. Fluids (1977) A turbulent closure should guarantee realizable conditions for the computed Reynolds stress tensor Renormalization group approach : RNG k t ε model (rather weak impact on turbulence closure) Yakhot, Orszag, Thangam, Gatski & Speziale (1992) 37 Airbus - EEA Faculty - Computational Aeroacoustics

38 Numerical simulation of turbulent flows The k t ε turbulence model Anomalously growth of k t near a stagnation point Contours of k t /U 2 (a) with realizability (b) without constraint Durbin, 1996, Int. J. Heat and Fluid Flow ν t = C µ k t k t ε T k ( t ε = min kt ε, 1 C µ 3 8 S 2 ) S 2 S ij S ij 38 Airbus - EEA Faculty - Computational Aeroacoustics

39 Numerical simulation of turbulent flows The k t ε turbulence model Behnia, Parneix & Durbin, Int. J. Heat Mass Transfer (1998) 39 Airbus - EEA Faculty - Computational Aeroacoustics

40 Numerical simulation of turbulent flows How interprete Unsteady RANS (URANS) simulations? Obtained through time-marching algorithms, t Ū + F(Ū) = 0 Mean shear flow Ū 1 = Ū 1 (x 2 ) Ū 2 = Ū 3 = 0 The Schwarz s inequality u 1 u 2 provides for the k t ε model, 2 u 2 1 u 2 2 9C 2 µ S 2 12 ε2 k 2 t frequency 2 Implicit low-pass filter imposed by the mean shear : no development of energy cascade with grid rafinement. The constant C µ (and thus ν t ) is reduced in practice, 3-D is required. Semi-deterministic modelling (Ha Minh, 1991) Bastin, Lafon & Candel, J. Fluid Mech. (1997) Spalart, Int. J. Heat and Fluid Flow (2000) Iaccarino et al. Int. Journal Heat Fluid Flow (2003) 40 Airbus - EEA Faculty - Computational Aeroacoustics

41 The k t ε model How interprete URANS simulations? Passive scalar in a subsonic plane jet (C µ ց) Bastin, Lafon & Candel, J. Fluid Mech. (1997) 41 Airbus - EEA Faculty - Computational Aeroacoustics

42 Numerical simulation of turbulent flows The k t ω SST (shear-stress transport) model Formal change of variable ε = C µ ωk t, and thus ν t = k t /ω, k t ε model transformed into a k t ω model dω dt = x j + ω k t x j [( µ + µ ) ] t ω + α P βρω 2 σ ε x j ν t [( µt µ ) ] t kt σ ε σ k x j + 2 k t (µ + σ ω µ t ) k t x j ω x j From standard k t ε model, α = C ε1 1, β = C µ (C ε2 1), σ ω = 1/σ ε, the transport term in gray is neglected. Without the cross-diffusion term in blue and specific values of α and β, Wilcox s model (1988, 2008, AIAA Journal) - better calibrated, no damping function near the wall - too sensitive to freestream values (Menter, 1991 ; Spalart & Rumsey, 2007 ; AIAA Journal ) 42 Airbus - EEA Faculty - Computational Aeroacoustics

43 Numerical simulation of turbulent flows The k t ω SST (shear-stress transport) model k t ω SST, one of the most widely used model Should be automatically used as default model in CFD codes Ad hoc modifications (constant calibration, F 1 blending function,...) Menter (1994) AIAA Journal dω dt = x j [( µ + µ ) ] t ω + α P βρω 2 ρ + 2(1 F 1 ) σ ε x j ν t σ ω2 ω k t x j ω x j Compressibility corrections Dezitter et al. AIAA Paper Rumsey J. Spacecraft & Rockets (2010) 43 Airbus - EEA Faculty - Computational Aeroacoustics

44 Numerical simulation of turbulent flows Statistical approaches (RANS) Supersonic underexpanded jet with flight effects M j = 1.35 (NPR = 2.97, convergent nozzle), M f = 0.4 elsa solver, k t ω SST (ONERA) Ū 1 k t Cyprien Henri, Ph.D. candidate, SNECMA (2011) SNECMA / AIRBUS 44 Airbus - EEA Faculty - Computational Aeroacoustics

45 Numerical simulation of turbulent flows The k t ω SST (shear-stress transport) model Dezitter et al., AIAA Paper (VITAL project) 45 Airbus - EEA Faculty - Computational Aeroacoustics

46 Numerical simulation of turbulent flows The k t ω SST (shear-stress transport) model Dezitter et al., AIAA Paper (VITAL project) 46 Airbus - EEA Faculty - Computational Aeroacoustics

47 Numerical simulation of turbulent flows Other strategies for solving RANS equations Reynolds stress models : RSM or R ij ε or R ij ε ij Not often used in industrial context Hanjalić & Launder (1972, 1976), Gatski & Speziale (1993, 1997, 2004), Gerolymos et al. Spalart-Allmaras (SA) model AIAA Paper , Recherche Aérospatiale (1994) transport equation on the turbulent viscosity d ν dt = c b 1 S ν + 1 σ { [(ν + ν) ν] + cb2 ( ν) 2} c w1 f w ( ν d w ) 2 ν related to the turbulent viscosity ν t = νf ν1 S vorticity magnitude sensor d w distance to the wall f w and f ν1 are damping functions (near-wall corrections) 47 Airbus - EEA Faculty - Computational Aeroacoustics

48 Large eddy simulation Closure : subgrid-scale model e.g. Sagaut (2006), Lesieur (2007) Projection modeling by a spacial convolution filtering ū = G u t ū + (ūū) + ( p/ρ) + ν 2 ū = σ sgs structural approach, find a model for the sgs stress tensor functional approach, surrogate the mean action turbulent kinetic energy balance is more important turbulent kinetic energy cascade is dominant dissipation Two main classes of methods in physical space turbulent eddy viscosity models hyperviscosity ( σ sgs ν h 2n ) or high-order explicit filtering Pruett et al. Domaradzki, Adams et al. Visbal, Gaitonde, Rizzetta ADM... LES - Relaxation filtering σ sgs = χg ũ Bogey & Bailly (2006), Berland et al., J. Comput. Phys. (2008) & JOT (2008) 48 Airbus - EEA Faculty - Computational Aeroacoustics

49 Compressible LES based on explicit filtering LES based on relaxation filtering (LES-RF) Filter shape & scale separation 4th-order filter 6th-order filter 8th-order filter by studying turbulence statistics, 4thorder still too dissipative LES having a same effective cut-off wavenumber provides similar results for sharp cut-off filters 10th-order filter Berland et al. (LaMSID UMR CNRS-EDF-CEA, Comput. Fluids, 2011) 49 Airbus - EEA Faculty - Computational Aeroacoustics

50 Large eddy simulation Closure : LES based on relaxation filtering (LES-RF) Spectrum model E(k) = C K (εl) 2/3 (kl) 5/3 f L (kl)f η (kl η ) L L large-scale length scale, l η Kolmogorov length scale Governing equation, t ū + (ūū) + ( p/ρ) + ν 2 ū = χg ũ Ẽ(k) = Ĝ 2 (k)e(k), Lin s equation for Ẽ t Ẽ(k) = T (k) 2νk 2 Ẽ(k) 2χ [ 1 Ĝ(k) ] Ẽ(k) D ν (k) = 2νk 2 Ẽ(k) viscous dissipation D f (k) = 2χ [ 1 Ĝ(k) ] Ẽ(k) dissipation induced by the explicit filtering Reynolds effects. Given Mach number, u = cst and k t = cst Reynolds number Re L = u L/ν L, and space discretization L/ = cst = For a given wavenumber k α k η, D f (k α ) u 3 cst D ν (k α ) u 3 Re 1 L 50 Airbus - EEA Faculty - Computational Aeroacoustics

51 Large eddy simulation Closure : LES based on relaxation filtering (LES-RF) k s c k g c = π/ Re L = 10 3 Re L = 10 5 E(k )/(u 2 L) E(k) energy spectrum k g c = π/ k s c = 2π/(5 ) k L/ = O(10) von Kármán (1948) - Pao (1965) E(k) = C K ε 2/3 k 5/3 f L (kl)f η (kl η ) 51 Airbus - EEA Faculty - Computational Aeroacoustics

52 Large eddy simulation Closure : LES based on relaxation filtering (LES-RF) E(k )/(u 2 L) D(k )/u ks c Re L ր k g c = π/ k Re L = 10 3 Re L = 10 5 E(k) energy spectrum D ν (k) viscous dissipation k g c = π/ k s c = 2π/(5 ) S(k) = k 0 (Bailly & Comte-Bellot, 2003) T (k )dk 0 for kλ g 1 = ks cλ g 1 for a well-resolved LES 52 Airbus - EEA Faculty - Computational Aeroacoustics

53 Large eddy simulation Closure : LES based on relaxation filtering (LES-RF) 10 0 E(k )/(u 2 L) D(k )/u Re L ր k s c Re L = 10 3 Re L = 10 5 Ẽ(k) energy spectrum D ν (k) viscous dissipation D f (k) explicit filtering π/1024 π/128 π/16 π/4 π k In the well-resolved wavenumber range, we must have D f (k) < D ν (k) constraint on the mesh size Regularization term corresponding to the net drain associated with the turbulent kinetic energy cascade, larger scales mostly unaffected 53 Airbus - EEA Faculty - Computational Aeroacoustics

54 Large eddy simulation Closure : LES based on eddy viscosity concept 10 0 E(k )/(u 2 L) D(k )/u k s c Re L ց as ν t ր Re L = 10 3 Re L = 10 5 E(k) energy spectrum D ν (k) viscous dissipation D ν+νt (k) eddy-viscosity (with ν t /ν = 5) π/1024 π/128 π/16 π/4 π k In the well-resolved wavenumber range, D ν (k) is replaced by D ν+νt (k), same functional form, overestimation of the dissipation for resolved scales However user-friendly for unstructured grids 54 Airbus - EEA Faculty - Computational Aeroacoustics

55 Numerical simulation of turbulent flows Status of large eddy simulation full-scale for laboratory jets (typically D = 2 cm, M = 0.9, Re D = ) and nearly mature numerical tool (basic statistics, turbulent kinetic energy budget, two-point space-time correlations, Reynolds effects) advances in «alternative subgrid-scale models» e.g. removing energy at the smallest resolved scale by explicit filtering with N 10 8 points, DNS at Re D 10 4 and LES at Re D 10 5, St Airbus - EEA Faculty - Computational Aeroacoustics

56 Numerical simulation of turbulent flows Direct Noise Computation around a NACA 0012 airfoil Chord-based Reynolds number Re c = Mach number M = 0.22 Span width of 5% of the chord, grid spacings in wall units close to the trailing edge : x + 20, y + 2.5, z + 20, mesh of points ω z vorticity around the trailing edge transition zone 0.6 x/c 0.75 snapshot of the fluctuating pressure field (color scales between ±5 Pa) Marsden et al., 2008, AIAA Journal, 46(4) ω x vorticity, blue and red surfaces correspond to ± s 1 56 Airbus - EEA Faculty - Computational Aeroacoustics

57 Numerical simulation of turbulent flows Direct Noise Computation of coplanar coaxial hot jets (CoJen) Velocities V p = m.s 1 and V s = m.s 1 Temperatures T sp = K and T ss = K AR = 3, VR = Reynolds number Re D2 = V s D 2 /ν = 10 6 D 2 = 4.9cm Grid of points & 400,000 time steps, T = 0.03s 1800h CPU Nec-SX5 Bogey et al., 2009, Phys. Fluids, 21 WEM at 60D 2 from LES results (LEE, pts) Pressure spectra at θ = 30 experimental data (QinetiQ, UK) snapshot of vorticity norm & temperature SPL (db/st) 10 db St r = 2D 2 r = 3D 2 r = 4D 2 57 Airbus - EEA Faculty - Computational Aeroacoustics

58 Numerical simulation of turbulent flows An unsteady Navier-Stokes simulation is necessarily 3-D Flow separation behind a rounded leading edge (Courtesy of Lamballais, Sylvestrini & Laizet, Int. Journal Heat Fluid Flow, 31, 2010) (η = 0.125) (η = 1) Spanwise vorticity ω z, from red to blue with ω z = ±5U /H, DNS with inflow perturbations u inflow = 0.1%U 2-D (no energy cascade / mixing) versus 3-D simulation Airbus - EEA Faculty - Computational Aeroacoustics

59 Numerical simulation of turbulent flows Hybdrid technique : Zonal Detached-Eddy Simulation (ZDES) Deck, Theoret. Comput. Fluid Dyn., 2011 Sagaut, Deck & Terracol, 2006, Multiscale and multiresolution approaches in turbulence 59 Airbus - EEA Faculty - Computational Aeroacoustics

60 Detached Eddy Simulation (DES) Numerical simulation of turbulent flows Approach combining the most favorable elements of RANS models and Large Eddy Simulation (LES) DES based on the Spalart-Allmaras RANS model d ν dt = c b 1 S ν + 1 σ { [(ν + ν) ν] + cb2 ( ν) 2} c w1 f w ( ν d) 2 SA-92, d = d w, the distance to the closest wall DES-97, d = min(d w, C DES ) ; DDES, d = min(d w, ΨC DES ) Turbulent flow at equilibrium (and high Re), ν t d 2 S Smagorinsky DES acts as RANS model (Spalart-Allmaras) when d (boundary layer), and as SGS-LES when d (separated regions) DES97 - Spalart, Jou, Streelets & Allmaras (1998) DDES - Spalart et al. (2006) Theoret. Comput. Fluid Dyn. Deck (2011) Theoret. Comput. Fluid Dyn. 60 Airbus - EEA Faculty - Computational Aeroacoustics

61 Numerical simulation of turbulent flows Lattice Bolzmann Method (LBM) Approximation of compressible Navier-Stokes equations (through simplified mesoscopic kinetic equations or discretization of Bolzmann equation) f(x, c, t) density distribution function ( number of particles at x and t with velocity c) Local macroscopic variables (pressure provided by an equation of state) ρ = fdc ρu = fcdc ρe = 1 (c u) 2 fdc 2 V (c) V (c) Without any collision, t f + c f = 0, but in general, f t + c f = Ω(f) V (c) Ω(f) collision operator Discretisation of the velocity space V (c) c α D3Q19-model 61 Airbus - EEA Faculty - Computational Aeroacoustics

62 Lattice Bolzmann Method (LBM) Numerical simulation of turbulent flows Robust numerical method, linear advection equation, ready for hpc (but requires hpc!), complex physics can be solved, simple mesh generation Interesting compressible formulation for low Mach number flows : direct computation of noise Open theoretical problems for high-speed flows and thermodynamics Turbulence modelling/closure : basically similar to Navier-Stokes equations (PowerFLOW - RNG model with ad hoc terms), second-order accuracy in both time and space, only basic boundary conditions currently available Chen & Doolen, Annu. Rev. Fluid Mech., 30, 1998 Ricot, Ph.D. thesis, ECL ( Malaspinas & Sagaut, Phys. Fluids, 23, Airbus - EEA Faculty - Computational Aeroacoustics

63 Numerical simulation of turbulent flows Landing gear AIAA Paper Seror et al., AIAA Paper Li et al. (PowerFLOW & RNG turbulence model) 63 Airbus - EEA Faculty - Computational Aeroacoustics

64 Numerical simulation of turbulent flows Helmholtz resonator excited by a grazing flow Resonator frequency response (L = 0.01 m, D = 0.08 m) Nelson, Halliwell & Doak s experiments, J. Sound Vib. (1981, 1983) Transfer function between the two probes B.E.M. (Sysnoise) PowerFLOW Fonction de transfert (db/hz) Frequence (Hz) Ref. Ricot, Maillard & Bailly, AIAA Paper & (PowerFLOW CFD code) 64 Airbus - EEA Faculty - Computational Aeroacoustics

65 Numerical simulation of turbulent flows Helmholtz resonator excited by a grazing flow Aerodynamic excitation SPL (db) Frequence (Hz) Vitesse (m/s) 200 Experiments : Helmholtz resonance f Hz U 12 m.s 1 Nelson, Halliwell & Doak s experiments, J. Sound Vib. (1981, 1983). See 2nd companion paper. Difficult to validate SPL Ref. Ricot, Maillard & Bailly, AIAA Paper & Airbus - EEA Faculty - Computational Aeroacoustics

66 From CFD to CAA : Formulations informed by steady / unsteady CFD 66 Airbus - EEA Faculty - Computational Aeroacoustics

67 Computational Aeroacoustics From CFD to CAA Formulations informed by steady / unsteady CFD Lighthill s theory of aerodynamic noise & variants Presence of solid boundaries Ffowcs-Williams & Hawkings CFD coupling Alternative approaches : APE, LEE, SNGR,... Steady CFD & statistical models applied to jet mixing noise Supersonic jet noise Mach waves & broadband shock-associated noise (BBSAN) Concluding remarks : strategies in CAA 67 Airbus - EEA Faculty - Computational Aeroacoustics

68 Computational Aeroacoustics Different levels of representation/modelling in aeroacoustics analogies hybrid methods unsteady DNC & WEM resolved physics & computational cost incompressible CFD compressible statistical models steady DNC = Direct Noise Computation WEM = Wave Extrapolation Methods 68 Airbus - EEA Faculty - Computational Aeroacoustics

69 Lighthill s theory of aerodynamic noise The simplest wave equation from the conservation of mass and Navier-Stokes equations (ρu i ) t ρ t + (ρu i) = 0 (1) x i + ( ρu i u j ) x j = p x i + τ ij x j (2) Sir James Lighthill ( ) t (1) (2) = 0 and c x 2 2 ρ = c 2 2 (ρδ ij ) i x i x j 2 ρ t 2 c2 2 ρ = 2 T ij x i x j with T ij = ρu i u j + ( p c 2 ρ ) δ ij τ ij Lighthill s tensor 69 Airbus - EEA Faculty - Computational Aeroacoustics Lighthill, Proc. Roy. Soc. London (1952) & AIAA Journal (1982)

70 Lighthill s theory of aerodynamic noise Interpretation of Lighthill s equation ρ = Λ tt c 2 2 Λ = T ρ Λ nonlinear effects In a uniform medium at rest ρ, p, c ρ = 0 ρ Λ turbulence ρ Λ mean flow effects 70 Airbus - EEA Faculty - Computational Aeroacoustics

71 Lighthill s theory of aerodynamic noise Retarded-time solution of Lighthill s equation ρ (x, t) = 1 4πc 4 V 2 T ij y i y j ( y, t r ) dy c r observer (x, t) By using r = x y x x y x +O 1 x i y i c x t ρ (x, t) 1 x i x j 4πc x 4 x 2 x y V 2 T ij t 2 in the far-field approximation ( ) y 2 x x y ( y, t r ) dy c r = x y x S y x = x y x y O source volume V of turbulence 71 Airbus - EEA Faculty - Computational Aeroacoustics

72 Lighthill s theory of aerodynamic noise Some remarks about these subtle integral formulations Crighton (1975), Ffowcs Williams (1992) May we neglect the retarded time differences in the integral solutions? t r x y = t c c t x c + x y xc + observer (x, t) difference in time emission x y xc l s c r = x y difference in time emission source turbulent time Yes if M t 1, compact sources (M t turbulent Mach number) l s/c l s /u M t y x S S y x y y O source volume V l 3 s of turbulence x 72 Airbus - EEA Faculty - Computational Aeroacoustics

73 Lighthill s theory of aerodynamic noise Insightful example (Crighton, 1974) p = S with S (y, τ) = ρ 0 u 2 l 2 s e y2 l 2 u2 τ 2 s l 2 s p (x, t) = ρ 0 c 2 0 π 4 l s x M 2 ( ) 2 e u2 (1 + M 2 l 2 t x 1 c s 0 1+M 2 x y ) 1/2 M 0 π p (x, t) = ρ 0 c0 2 4 l s x M2 e u 2 ( ) 2 l 2 t c x s 0 M p (x, t) = ρ 0 c 2 0 π 4 ( ) 2 l sx M e c2 0 l 2 t c x s 0 W M 4 W M 2 source frequency u/l s acoustic frequency u/l s source frequency u/l s acoustic frequency c 0 /l s 73 Airbus - EEA Faculty - Computational Aeroacoustics

74 Lighthill s theory of aerodynamic noise Simplification of the source term T ij Viscous effects as noise sources are negligible, T ij ρu i u j + ( p c 2 ρ ) δ ij Moreover, p = c 2 ρ + (p /c v ) s for a perfect gas. Hence for flows nearly isentropic, T ij ρu i u j For low Mach number isothermal flows T ij ρu i u j ρ u i u j... but acoustic - mean flow interactions are definitively lost Mean flow effects are contained in the linear compressible part of the Lighthill tensor T ij Aerodynamic noise source term non-linear part of T ij The integral solution is a convolution product. In free space, ρ = G 2 T ij x i x j = 2 G x i x j T ij = 2 ( ) G Tij x i x j 74 Airbus - EEA Faculty - Computational Aeroacoustics

75 Other formulations derived from Lighthill s analogy Vortex sound theory Powell (1964), Howe (1975), Möhring (1978), Yates (1978)... Reformulation of Lighthill s equation to emphasize the role of vorticity (localized quantity) in the production of sound For incompressible flows, u = 0, at low Mach number, ( ) u (uu) = (ω u) L = ω u Lamb s vector 2 ρ t 2 c2 2 ρ ρ (ω u) 75 Airbus - EEA Faculty - Computational Aeroacoustics

76 Numerical implementation of integral formulations Source term formulation (space / temporal derivatives / Fourier s space) Ref. Sarkar & Hussaini, ICASE (1993) Bastin, Lafon & Candel, J. Fluid Mech., 335 (1997) Lockard, J. Sound Vib., 229(4) (2002) Gloerfelt, Bailly & Juvé, J. Sound Vib., 266(1) (2003) Algorithm for retarded-time (time accumulation / source-time algorithm) Ref. Sarkar & Hussaini, ICASE (1993) Brentner & Farassat, Prog. in Aero. Sci., 39 (2003) Casalino, J. Sound Vib., 261 (2003) Truncation of the source volume Ref. Witkowska & Juvé, C. R. Acad. Sci. Paris, 318 (1994) Wang, Lele & Moin, AIAA Journal, 34(11) (1996) Bastin, Lafon & Candel, J. Fluid Mech. (1997) FEM, FMM Ref. Oberai, Ronaldki & Hughes Comput. Methods Appl. Mech. Engrg., 190 (2000) Wolf & Lele, AIAA Journal, 49(7) (2011) 76 Airbus - EEA Faculty - Computational Aeroacoustics

77 Ffowcs Williams & Hawkings formulation Presence of solid (moving) boundaries ρ (x, t)? n = f/ f turbulent flow f < 0 Σ/f = 0 volume V, f > 0, closed by Σ Generalized function H(f)ρ variable valid over all space, (Hρ)? 2 Hρ t 2 c 2 2 Hρ = 2 HT ij + F iδ (f) x i x i x i x j x i + Qδ (f) t T ij = ρu i u j + ( p c 2 ρ ) δ ij τ ij F i = [ ρu i (u j u Σ j ) + pδ ij τ ij ] f x j (Lighthill s tensor) Q = [ ρ(u j u Σ j ) + ρ u Σ j ] f x j Ffowcs Williams & Hawkings, Phil. Trans. Roy. Soc. London, (1969) 77 Airbus - EEA Faculty - Computational Aeroacoustics

78 Ffowcs Williams & Hawkings formulation Integral solution Hρ (x, t) = 1 4πc πc 2 2 x i x j x i Σ 1 r V T ij (y, τ ) dy 1 M r r F i (y, τ ) 1 M r dσ f + 1 4πc 2 M r = u Σ r c r t Σ 1 r Q (y, τ ) 1 M r τ = t r c dσ f volume integral volume outside the surface controle! 2 surface integrals controle surface Lighthill s tensor, quadrupole term. loading noise, dipole term. thickness noise, monopole term. Helicopter noise applications : Farassat, Brentner, Myers, Prieur, Airbus - EEA Faculty - Computational Aeroacoustics

79 Ffowcs Williams & Hawkings formulation Two classical configurations for the surface control cavity noise cylinder noise Σ/f = 0 Σ/f = 0 U U f < 0 V 0 S 0 f < 0 S 0 V 0 79 Airbus - EEA Faculty - Computational Aeroacoustics

80 Curle s formulation Non-porous surface u j u Σ j at rest u Σ j = 0 (Q 0) Hρ (x, t) = 1 4πc 2 2 x i x j 1 x i x j 4πc x 4 x 2 V V T ij (y, τ ) dy r 1 4πc 2 x i 2 T ij t (y, 2 τ ) dy + 1 x i 4πc x 3 x Σ p ij (y, τ ) n jdσ r Σ p ij t (y, τ ) n j dσ in the acoustic far-field, with p ij = pδ ij τ ij Compact surface retarded time negligeable if L Σ λ Hρ (x, t) 1 4πc x 3 t Σ p ( y, t x ) cos θdσ c fluctuating force for a compact surface, equivalent dipole source dimensionnal analysis, W U 6 power law L Σ Σ θ n r = x y V 80 Airbus - EEA Faculty - Computational Aeroacoustics

81 Flow induced cylinder noise 2-D cylinder D = 3.81 cm M = 0.12 Re D URANS simulation (Fluent, RSM, two-layer method for BL) physical domain [ 8.5D; 16.5D] [ 10.5D; 10.5D] pts, 0.3 y on the cylinder circumference, t 10 4 s initial free-stream turbulence level around 1% integration in the frequency domain (over two periods) for acoustics 1.5 x 2 /D x 1 /D Vorticity snapshot, contours 16 between ωd/u = ±4.43 Ref. Pérot, F. (Ph.D. Thesis, PSA Peugeot-Citroën) Gloerfelt, Pérot, Bailly & Juvé, J. Sound Vib., 287, Airbus - EEA Faculty - Computational Aeroacoustics

82 Flow induced cylinder noise Mean pressure coeffcient C p Drag and lift coeffcients C p C L C D θ simulation at Re D = C D 0.47 turbulent stream uniform stream (Re D = , data from Batham, 1973) t/t St = f 0D 0.24 U C L = 1 2π p sin θdθ f ρ U C D = 1 ρ U 2 2π 0 p cos θdθ 2f 0 82 Airbus - EEA Faculty - Computational Aeroacoustics

83 Flow induced cylinder noise Curle s analogy H(f) p(x, ω) = V T ij 2 Ĝ 0 y i y j dy Σ p Ĝ0 y i n i dσ Ĝ 0 (x y, ω) = 1 + 4i H(1) 0 (kr) φ(x, ω) = φ(x, t)e iωt dt Tailored Green s function Ĝ1 H(f) p(x, ω) = V T ij 2 Ĝ 1 y i y j dy Σ p Ĝ1 y i n i dσ(y) =0 Ĝ 1 = Ĝ0 + Ĝs incident + scattered Σ p Ĝ0 y i n i dσ = V T ij 2 Ĝ s y i y j dy 83 Airbus - EEA Faculty - Computational Aeroacoustics

84 Flow induced cylinder noise Curle s analogy Surface integral Volume integral pressure levels between ±5 Pa dipole-like field at frequency f 0 pressure levels between ±0.5 Pa lateral quadrupole field at f 0 84 Airbus - EEA Faculty - Computational Aeroacoustics

85 Flow induced cylinder noise Curle s analogy Surface integral x 2 component (lift) Surface integral x 1 component (drag) pressure levels between ±5 Pa dipole-like field at frequency f 0 pressure levels between ±0.5 Pa dipole-like field at 2f 0 85 Airbus - EEA Faculty - Computational Aeroacoustics

86 Flow induced cylinder noise Tailored Green function Gloerfelt et al. J. Sound Vib., 287, 2005 Contribution of G s Contribution of G 0 pressure levels between ±5 Pa dipole-like field at frequency f 0 pressure levels between ±0.5 Pa lateral quadrupole field at f 0 aerodynamic sources located at x 1 1.5D behind the cylinder producing a quadrupole field at f 0, whose scattered field is a dipole field at f 0. Curle surface integral scattering of aerodynamic sources by the cylinder 86 Airbus - EEA Faculty - Computational Aeroacoustics

87 Wave Extrapolation Methods Methods for extending near-field to far-field Kirchhoff s integral theorem Ffowcs Williams & Hawkings (porous or permeable surface) Prieur & Rahier, 2001, Aerosp. Sci. Technol. (addendum in vol. 6) Morfey & Wright, 2007, Proc. Roy. Soc. London Spalart & Shur, 2009, Int. Journal of Aeroacoustics Linearized Euler s Equations Weakly Non-Linear Euler Equations (WNLEE) Gloerfelt, Bogey & Bailly, 2003, Int. Journal of Aeroacoustics (Full) Euler s equations de Cacqueray et al., 2011, AIAA Journal & AIAA Paper FFWH - cavity noise Gloerfelt et al. (2003) J. Sound Vib. 266(1) 87 Airbus - EEA Faculty - Computational Aeroacoustics

88 Wave Extrapolation Methods Application of integral methods to cavity noise Acoustic analogy Ffowcs Williams & Hawkings Wave Extrapolation Methods Kirchhoff s method Ffowcs Williams & Hawkings (based on / porous FW-H) Gloerfelt, Bailly & Juvé, 2003, J. Sound Vib., 266(1) 88 Airbus - EEA Faculty - Computational Aeroacoustics

89 Wave Extrapolation Methods Ffowcs Williams & Hawkings analogy 2-D formulation in the frequency-domain φ (x, ω) = + φ (x, t) e iωt dt with a uniform mean flow in observer region, p(x, ω) = Fi (y, ω) G conv L 0 y i with in this case U observer x dσ(y) Tij (y, ω) 2 G conv dy S 0 y i y j O r = x y y S 0 L 0 n volume source + cavity rigid walls [u n = 0] T ij = ρ(u i U i )(u j U j ) + ( p c 2 ρ ) δ ij τ ij F i = ( pδ ij τ ij ) nj G conv (x, y, ω) = i 4β ei(mk(x 1 y 1 )/β 2) H (2) 0 ( ) krβ β 2 89 Airbus - EEA Faculty - Computational Aeroacoustics

90 Wave Extrapolation Methods Ffowcs Williams & Hawkings analogy volume integral surface integrals FW-H analogy DNS reference solution separation between direct and reflected sound fields (Powell reflection theorem, 1960) volume integral sensitive to truncature effects 90 Airbus - EEA Faculty - Computational Aeroacoustics

91 Wave Extrapolation Methods Wave extrapolation methods 2-D formulation in the frequency-domain φ (x, ω) = + φ (x, t) e iωt dt with a uniform mean flow in observer region, U observer x O r = x y y porous surface [u n 0] Ffowcs Williams & Hawkings WEM (porous FW-H) p(x, ω) = F i (y, ω) G conv dσ(y) iω Q(y, ω) G conv dσ(y) y i L L n L F i = [ ρ(u i 2U i )u j + pδ ij τ ij ] nj Q = ρu i n i Convected Kirchhoff method 91 Airbus - EEA Faculty - Computational Aeroacoustics

92 Wave Extrapolation Methods Ffowcs Williams & Hawkings WEM DNS reference solution FW-H WEM from L 1 FW-H WEM from L 2 FW-H WEM from L 3 92 Airbus - EEA Faculty - Computational Aeroacoustics

93 Wave Extrapolation Methods Convected Kirchhoff method DNS reference solution Kirchhoff from L 1 Kirchhoff from L 2 Kirchhoff from L 3 93 Airbus - EEA Faculty - Computational Aeroacoustics

94 Mean flow effects Quiz Flow past a cavity : observer in a uniform medium U, ρ, c How to interpret Lighthill s equation? Introducing the following (arbitrary) decomposition, ρ ρ + ρ u i U δ 1i + u i U + T ij = ρu i u j = (ρ + ρ )(U δ 1i + u i)(u δ 1j + u j) Lighthill s equation can be rearranged as, Gloerfelt et al., 2003, J. Sound Vib., ρ t 2 c2 2 ρ = 2 (ρu x i x iu 2 j) + 2U (ρu j x 1 x j) + U 2 2 ρ j x1 2 = 2 (ρu x i x iu 2 ρ j) 2U U 2 2 ρ j t x 1 x1 2 by using the conservation of mass ρ t + U ρ + (ρu x 1 x j) = 0 j 94 Airbus - EEA Faculty - Computational Aeroacoustics

95 Mean flow effects Quiz How to interpret Lighthill s equation? + In a uniform medium U, ρ, c, sound is governed by the convected wave equation, U ( ) 2 t + U ρ c x 2 2 ρ = 0 1 ( ) 2 t + U ρ 2 ρ x 1 t + 2U 2 ρ 2 + U 2 2 ρ t x 1 x1 2 Gloerfelt et al., 2003, J. Sound Vib., 266 Lighthill 2 ρ t 2 c2 2 ρ = 2 (ρu x i x iu 2 ρ j) 2U U 2 2 ρ j t x 1 x1 2 Mean flow - acoustic interactions are included in T ij Aerodynamic noise source term non-linear part of T ij 95 Airbus - EEA Faculty - Computational Aeroacoustics

96 Mean flow effects The Linearized Euler Equations (LEE) Small perturbations arround a steady mean flow ( ρ, ū, p) (no gravity) t ρ + (ρ ū + ρu ) = 0 t ( ρu ) + ( ρūu ) + p + ( ρu + ρ ū ) ū = 0 t p + [p ū + γ pu ] + (γ 1) p ū (γ 1) u p = 0 Acoustic propagation in the presence of a flow (atmosphere, ocean, turbulent flow,...) is governed by LEE In the general case, this system cannot be reduced exactly to a single wave equation. Blokhintzev (1946) Pridmore-Brown (1958), Lilley (1972), Goldstein (1976, 2001, 2003) 96 Airbus - EEA Faculty - Computational Aeroacoustics

97 Mean flow effects The Linearized Euler Equations (LEE) For a parallel base flow ū i = ū 1 (x 2, x 3 )δ 1i, ρ = ρ(x 2, x 3 ) (and thus p = p constant), the LEE can be recasted into a wave equation based on the pressure, L 0 D Dt L 0 [p ] = 0 [ D2 Dt ( c 2 ) ] + 2 c 2 ū 1 2 i = 2, 3 D t + ū 1 2 x1 x i x 1 x i From the (exact) Navier-Stokes equations, we can also form an inhomogeneous wave equation based on L L 0 at leading order, L [p ] = Λ Lilley (1972) { d d 2 π dt dt ( c 2 π )} + 2 u i 2 x i x i x j +2 u i x j x i ( c 2 π ) = 2 u i u j u k x j x j x k x i x i ( ) ( ) 1 τ ij + d2 1 ds d { ( 1 ρ x i dt 2 c p dt dt x i ρ )} τ ij x j π = ln p π (1/γ)p /p 97 Airbus - EEA Faculty - Computational Aeroacoustics

98 Mean flow effects Lilley s equation Seek a solution of the form p (x, t) = φ (x 2 ) e i(k 1x 1 ωt) = φ (x 2 ) e ik(ν 1x 1 c o t) k = ω/c, ν 1 = k 1 /k = cos θ, M = ū 1 /c shear-layer thickness δ φ + 2ν 1 dm [ ] φ + k 2 (1 ν 1 M) 2 ν1 2 φ = 0 1 ν 1 M dx 2 k 2 k/δ k 2 High-frequency approximation, φ + q(x 2 )φ = 0 with q(x 2 ) = [ ] (1 ν 1 M) 2 ν1 2 q(x 2 ) < 0 exponential decrease q(x 2 ) > 0 periodic oscillations Turning point given by q(x 2 ) = 0, cos θ = 1/(M + 1) 98 Airbus - EEA Faculty - Computational Aeroacoustics

99 Mean flow effects Lilley s equation Harmonic source in a Bickley jet ū 1 u j = 1 cosh 2 (βy/δ) β = ln(1 + 2) St = 4.4 M = 0.5 λ δ LEE (log 10 ( p + ε)) and ray-tracing high-frequency noise is diverting away from the jet axis shadow zone at angles close to the jet axis, θ 48.2 o (edge of the silence cone) 99 Airbus - EEA Faculty - Computational Aeroacoustics

100 Mean flow effects High-frequency solution Wavelength matching at the interface, c 1 f ( ) 1 + M 1 cos θ 1 = c 2 f 1 cos θ 2 x 2 p t 2 1 ū 1 θ 1 θ 2 p i p r θ 1 x cos θ 2 = cos θ M 1 cos θ 1 (0 θ 1 π) θ cos θ 2 = M 1 for θ 1 = 0 o (grazing incidence) θ 1 θ 2 = f(θ 1 ) for M 1 = 0 : 0.1 : Airbus - EEA Faculty - Computational Aeroacoustics

101 Linearized Euler s Equations LEE not so well-posed as wave operator Radiation and refraction of sound waves through a 2-D shear layer (4th CAA workshop, NASA CP ) x 2 PML (Hu, 2001) ρ(x 2 ), ū 1 (x 2 ) O S 1 x 1 sound field instability wave (Kelvin-Helmholtz for this case) Thomas Emmert Diplomarbeit Technische Universtät München - ECL 101 Airbus - EEA Faculty - Computational Aeroacoustics

102 Linearized Euler s Equations L 0 [p ] = 0 L 0 LEE generalization of the Rayleigh equation (1880) for a compressible perturbation in inviscid stability theory three families of instability waves (including Kelvin-Helmholtz) which can overwhelm the acoustic solution or/and generate noise (dominant noise mechanism for supersonic jets) Tam & Burton, 1984, J. Fluid Mech. Tam & Hu, 1989, J. Fluid Mech. Tam, 1995, Annu. Rev. Fluid Mech. acoustic and instability waves are coupled except, - for the high-frequency limit, geometrical acoustics (ray tracing) - for a potential based mean flow interesting numerical test case in the proceedings of the 4th CAA workshop Agarwal, Morris & Mani, 2004, AIAA Journal, 42(1) 4th CAA workshop, NASA CP Airbus - EEA Faculty - Computational Aeroacoustics

103 Linearized Euler s Equations Alternative approaches LEE in the frequency domain Ref. Rao & Morris, 2006, AIAA Journal, 44(7) Karabasov, Hynes & Dowling, 2007, AIAA Paper «Simplified» formulation of LEE (by removing a gradient term associated with refraction effects) & hybrid method using source terms in LEE Ref. Bogey, Bailly & Juvé, 2002, AIAA Journal, 40(2) & AIAA Paper Bogey, Gloerfelt, Bailly, 2003, AIAA Journal, 41(8) Bailly & Bogey, 2004, IJCFD, 18(6) Bailly, Bogey & Candel, 2010, Int. J. Aerocoustics Colonius, Lele & Moin, 1997, J. Fluid Mech. Goldstein, 2001, 2003, 2005, J. Fluid Mech. 103 Airbus - EEA Faculty - Computational Aeroacoustics

104 Linearized Euler s Equations Alternative approaches (con d) Acoustic Perturbation Equations (APE), sound propagation in an irrotational mean flow (extension of vortex sound theory) t ρ + xi (ρ ū i + ρu i ) = 0 t u i + xj (ū i u j) + xi (p / ρ) = q m t p c 2 t ρ = 0 q m (ω u) Ref. Howe (1998), Möhring (1999) Ewert & Schröder, 2003, J. Comput. Phys. «Galbrun s school» 104 Airbus - EEA Faculty - Computational Aeroacoustics

105 Linearized Euler s Equations «Simplified» formulation of LEE by removing a gradient term associated to refraction effects 6 x M j = T j = 600 K St = f 0 2b/u j Bailly & Bogey, NASA CP pressure along the line y = 15b analytical solution Agarwal, Morris & Mani (AIAA Journal, 2004) LEE approximation not valid for low frequency applications classical high-frequency approximation (geometrical acoustics) 105 Airbus - EEA Faculty - Computational Aeroacoustics

106 Computational Aeroacoustics From CFD to CAA Formulations informed by steady / unsteady CFD Lighthill s theory of aerodynamic noise & variants Presence of solid boundaries Ffowcs-Williams & Hawkings CFD coupling Alternative approaches : APE, LEE, SNGR,... Steady CFD & statistical models applied to jet mixing noise Supersonic jet noise Mach waves & broadband shock-associated noise (BBSAN) Concluding remarks : strategies in CAA 106 Airbus - EEA Faculty - Computational Aeroacoustics

107 Statistical modelling of mixing noise Lightill s theory When a time-dependent solution of T ij is not available, an alternative approach is to estimate the autocorrelation function of the acoustic pressure defined as, R a (x, τ) = p (x, t)p (x, t + τ)/(ρ c ) Acoustic intensity, I(x) = R(x, τ = 0) Power spectral density, S a (x, ω) = 1 2π + R a (x, τ)e iωτ dτ Turbulence statistically stationary, R a (x, τ) = 1 x i x j x k x l 16π 2 ρ c x 5 2 x 4 V V 4 τ 4T ij[y A, t] T kl [y B, t + τ] dy A dy B R ijkl ( y, η, τ + τη ) fourth-order two-point two-time correlation tensor y = y A η = y B y A τ η = x η/(xc ) 107 Airbus - EEA Faculty - Computational Aeroacoustics

108 Statistical modelling of mixing noise Lightill s theory R a (x, τ) = 1 x i x j x k x l 16π 2 ρ c x 5 2 x 4 V { V 4 } τ R ( ) 4 ijkl y, η, τ + τη dη dy moving frame to separate the convective amplification from the evolution of the turbulence itself (Doppler or convection factor) isotropic turbulence locally over the η integration two turbulence scales needed, e.g. with the k t ε turbulence model, L k3/2 t ε τ k t ε... two-point time correlation function? Ref. MGB model (Mani, Gliebe, Balsa) & Kharavan, AIAA J., 37(7), 1999 Bailly, Lafon & Candel, J. Sound Vib., 194(2), 1996 & AIAA J., 35(11), 1997 Tam & Auriault AIAA J., 37(2), 1999 & AIAA J. 42(1), 2004 Morris & Farassat AIAA J., 40(4), Airbus - EEA Faculty - Computational Aeroacoustics

109 Subsonic turbulent jet flow Space-time velocity correlations by dual-piv (Fleury et al., AIAA Journal, 2008) Re D = , M = 0.9, D = 3.8 cm, δ θ /D init At x = 5D, L (1) D, Kolmogorov scale l η 10 4 D Space-time second-order correlation functions R 11 (x, ξ, τ) and R 22 (x, ξ, τ) measured at x = (6.5D, 0.5D) L (1) 11 2δ θ L (1) 22 δ θ ξ 2 D ξ 2 D τ = 0 µs τ = 50 µs τ = 150 µs τ = 250 µs ξ 1 /D ξ 1 /D ξ 2 D ξ 2 D ξ 1 /D ξ 1 /D ξ 2 D ξ 2 D ξ 1 /D ξ 1 /D ξ 2 D ξ 2 D ξ 1 /D ξ 1 /D 109 Airbus - EEA Faculty - Computational Aeroacoustics

110 Statistical modelling of mixing noise Lightill s theory A key result : to radiate sound, turbulence must have a sonic phase velocity in the observer direction Wavenumber frequency spectrum H ijkl (y, k, ω) = 1 (2π) 4 V + R ijkl (y, η, τ) e i(ωτ k η) dηdτ Power spectral density S a (x, ω) = π 2 ω 4 x i x j x k x l ρ c x 5 2 x 4 V ( H ijkl y, ω ) x c x, ω dy Condition for sound radiation, k turbulence = ω x c x 110 Airbus - EEA Faculty - Computational Aeroacoustics

111 Statistical modelling of mixing noise Subsonic round jets JEAN European program, Ph.D. Thesis G. Bodard / SNECMA (2009) RANS with a k t ε turbulence model Isothermal jets at M = 0.75 and M = 0.9 Cedre solver (ONERA), nodes, structured hexahedral mesh 111 Airbus - EEA Faculty - Computational Aeroacoustics

112 Statistical modelling of mixing noise Subsonic round jets JEAN European program, Ph.D. Thesis G. Bodard / SNECMA (2009) Acoustic spectra at r = 30D θ = 90 o θ = 60 o 10 db St St M = 0.75, T j /T = 1 M = 0.9, T j /T = 1 M = 0.9, T j /T = 2 simplified Tam & Auriault model 112 Airbus - EEA Faculty - Computational Aeroacoustics

113 Statistical modelling of mixing noise Critical analysis : the devil is in the details! Mean flow effects : numerical computation of the Green function Generalization to more complex flow : co-axial jets, flight effects, noise reduction devices,... Fidelity of RANS turbulence models, calibration of the acoustic model, correlation tensor R ijkl (y, η, τ) 113 Airbus - EEA Faculty - Computational Aeroacoustics

114 Work done by the instructors! DNC of subsonic mixing noise by LES Bogey, C., Bailly, C. & Juvé, D., 2003, Theoret. Comput. Fluid Dyn., 16(4), Bogey, C. & Bailly, C., 2006, Theoret. Comput. Fluid Dyn., 20(1), Bogey, C. & Bailly, C., 2006, Phys. Fluids, 18, , 1-14 Bogey, C. & Bailly, C., 2006, Comput. & Fluids, 35(10), Bogey, C. & Bailly, C., 2007, J. Fluid Mech., 583, Bogey, C., Barré, S. & Bailly, C., 2008, Int. J. Aeroacoustics, 7(1), 1-22 Bogey, C., Barré, S., Juvé, D. & Bailly, C., 2009, Phys. Fluids, 21, , 1-14 Bogey, C. & Bailly, C., 2010, J. Fluid Mech., 663, Bogey, C., Marsden, O. & Bailly, C., 2011, Phys. Fluids, 23, , 1-20 & 23, Bogey, C., Marsden, O. & Bailly, C., 2012, J. Fluid Mech., 701, Bogey, C., Marsden, O. & Bailly, C., 2012, Phys. Fluids, 24, Airbus - EEA Faculty - Computational Aeroacoustics

115 Subsonic turbulent jets Initial conditions at the nozzle exit (visualizations by T. Castelain & B. André, ECL) Re D Re D Re D Re D fully laminar u e/u j < 1% nominally laminar u e/u j 1% Re D 10 5 (Re δθ 300) transitional jets nominally turbulent u e/u j 10% Re D fully turbulent Re D Re D = u j D/ν Re δθ = u j δ θ /ν σ ue = u e/u j 115 Airbus - EEA Faculty - Computational Aeroacoustics

116 Tripped subsonic round jets Influence of initial turbulence level M = 0.9, Re D = 10 5, δ θ /r 0 = 1.8%, Re δθ = 900 σ ue = 0% σ ue = 3% σ ue = 6% n r n θ = n z = = 252 million pts as the exit turbulence level increases, coherent structures (and consequently vortex rolling-ups and pairings) gradually disappear σ ue = 9% higher initial turbulence levels lead to longer potential cores σ ue = 12% 116 Airbus - EEA Faculty - Computational Aeroacoustics

117 Tripped subsonic round jets Influence of initial turbulence level M = 0.9, Re D = 10 5, δ θ /r 0 = 1.8%, Re δθ = 900 <u z u z > 1/2 /u j z/r 0 u z/u j along r = r 0 σ ue = 0% σ ue = 3% σ ue = 6% σ ue = 9% σ ue = 12% M = 0.9 & Re D = Fleury et al., AIAA Journal (2008) As the initial turbulence level increases, the shear layers develop more slowly with lower rms velocity peaks (overshoot around the pairing position for u e/u j 6%, but nearly monotonical growth for u e/u j 9%) 117 Airbus - EEA Faculty - Computational Aeroacoustics

118 Tripped subsonic round jets Influence of initial turbulence level M = 0.9, Re D = 10 5, δ θ /r 0 = 1.8%, Re δθ = 900 σ e = 0% σ e = 3% σ e = 6% σ e = 9% σ e = 12% z = r 0 snapshots of vorticity ω z (r, θ) z = 2r 0 z = 4r Airbus - EEA Faculty - Computational Aeroacoustics

119 Supersonic jet noise Direct link between large scale structures and the radiated noise Large scale structures can be described by instability waves, well-established theoretically and experimentally Mach wave radiation cos θ = c /U c = 1/M c wave front c θ U c Strioscopy of an underexpanded supersonic jet (convergent nozzle), NPR = 5, T t = 293 K, D = 22 mm, exposure time 20 ns. wavy wall (instability wave) Courtesy of ONERA DMAE Airbus - EEA Faculty - Computational Aeroacoustics

120 Supersonic jet noise Radiation of instability waves Tam & Morris (1980), Tam & Burton (1984), Tam & Hu (1990) Matching to the near acoustic field (dimensionless variables) + ( p (z, r, θ, t) = ĝ (ξ) H n (1) iλξ r ) e i(ξz+nθ ωt) dξ λ ξ = ξ 2 ρ Mj 2ω2 ĝ (ξ) = 1 + { z A 0 exp i k(ω, z ) dz }e iξz dz 2π 0 where k is provided by the (inviscid) stability theory Far-field radiation for ξ ξ c, peak noise angle cos θ p ξ peak /(ρ 1/2 M j ω) r R e [g(z)] ĝ(ξ) subsonic O z Mach waves phase velocity v φ 0 ξ c = ρ 1/2M jω ξ 120 Airbus - EEA Faculty - Computational Aeroacoustics

121 Supersonic jet noise Radiation of instability waves by solving linearized Euler s equations Eggers (1966), 3-D jet, M= 2.22, T j /T, St=0.6/π, n = 1 Sprint 3-D - Bailly (2004) ONERA - ECL, Piot et al., Int. J. Aeroacoustics (2006) 121 Airbus - EEA Faculty - Computational Aeroacoustics

122 Supersonic jet noise Broadband shock-associated noise (BBSAN) Interaction between shock-cell structure and instability waves (Tam et al.) traveling wave e i(αx ωt), α ω/u c shock-cell structure cos(k s x), k s 2π/L s (L s mean shock cell spacing) interaction [ e i(α k s)x + e i(α+k s)x ] e iωt Component with supersonic phase velocity v φ = ω/(α k s ) (condition for sound radiation) cos(ψ) = c = c v φ ω (k s α) = c ( 2π ω ) = c c ω L s u c fl s u c f u c L s [1 + (u c /c ) cos ψ] (θ = π ψ) constructive scattering of instability waves / large scales by the quasiperiodic shock-cells in the jet plume v φ ψ c θ x 122 Airbus - EEA Faculty - Computational Aeroacoustics

123 Supersonic jet noise Narrow-band acoustic spectra in db/st 10 db θ = 30 NPR = 3.68 M j = 1.50 M d = 1 r = 53.2D p DSP (db/st) θ = 50 θ = 70 θ = 90 André et al. (2011) harmonics of screech tone θ = 110 θ = 130 St e = Tam s model for BBSAN u c (D e /u e ) L s (1 M c cos θ) 90 θ = St 123 Airbus - EEA Faculty - Computational Aeroacoustics

124 Discussion : jet noise reduction, recent developments 124 Airbus - EEA Faculty - Computational Aeroacoustics

125 Jet noise reduction Promoting mixing... but it does not automatically lead to noise reduction! High-bypass-ratio nozzle (cfm56 type) chevrons on the fan and core nozzles (Loheac et al., SNECMA, 2004) QTD2 - Boeing - NASA AIAA Paper Castelain et al. AIAA Journal, 2008, 45(5) Saiyed et al., J. E., 2003, AIAA Journal Loheac et al., AIAA Paper Callender et al., 2005 & 2008, AIAA Journal 125 Airbus - EEA Faculty - Computational Aeroacoustics

126 Jet noise reduction Vortex generators : interpretation jet axis nozzle delta-tab Sketch of the formation of a pair of counter-rotating streawise vortices from a single delta or triangular tab mounted on a nozzle, and front view of vorticity field in a cross-section of the jet flow. Samimy et al. (1993), Zaman et al. (1994) 126 Airbus - EEA Faculty - Computational Aeroacoustics

127 Jet noise reduction Variable geometry or smart chevrons Calkins et al. (2006) «ideal scenario» Static chevrons used on the core nozzle to reduce cabin noise induced by shock cell structure during cruise conditions without thrust penalty, Smart chevrons only immersed into the fan flow during take-off for preserving airport community, and then retracted for thrust performance. 127 Airbus - EEA Faculty - Computational Aeroacoustics

128 Fan chevron versus core chevron Experimental study by SNECMA at CEPRA 19 (2010) Jet noise reduction (BPR = 9) decrease of low-frequency noise component, but penalty with fan-chevrons in high-frequency range ; balance : gain of EPNdB penalty for the nozzle thrust coefficient, C T 0.25% 0.30% shock-cell noise (cabin noise) reduced in cruise conditions with secondary chevrons 128 Airbus - EEA Faculty - Computational Aeroacoustics

129 Jet noise reduction B787-8 with chevrons Boeing 787-8, Trent 1000 / GEnx (Bourget Air Show, June 2011, B. André) 129 Airbus - EEA Faculty - Computational Aeroacoustics

130 Jet noise reduction Internal mixer Close-up of a CFM56-5C engine powering the Airbus A340 airliner. The fan diameter is 1.85 m, the bypass ratio is about 6.5, and a lobed exhaust ejector / mixer system is used to reduce the core jet speed inside the duct nacelle. Courtesy of Terence Li, photographer (2008). 130 Airbus - EEA Faculty - Computational Aeroacoustics

131 Concluding remarks : strategies in CAA 131 Airbus - EEA Faculty - Computational Aeroacoustics

132 Computational Aeroacoustics What could be a philosophy to use (often heavy) aeroacoustics simulations? real life concepts theory predict aerodynamic noise in pratical / realistic configurations provide the bounds of achievable noise reduction use CAA as diagnostic tool for studying specific (small) problems known-how : industrial softwares, hpc, reduced-order models understand physics of aerodynamic noise generation 132 Airbus - EEA Faculty - Computational Aeroacoustics

133 Computational Aeroacoustics Different levels of representation/modelling in aeroacoustics analogies hybrid methods unsteady DNC & WEM resolved physics & computational cost incompressible CFD compressible statistical models steady DNC = Direct Noise Computation WEM = Wave Extrapolation Methods 133 Airbus - EEA Faculty - Computational Aeroacoustics

134 Concluding remarks Turbulence models (RANS) Well-known behaviour of standard models Two classical models in aeronautics SA - ν t & k t ω SST Interpretation of URANS with caution LES status DES Advances in alternative subgrid-scale models LES - Relaxation Filtering : removing energy at the smallest resolved scale Nearly mature numerical tool - fidelity (basic statistics, turbulent kinetic energy budget, two-point space-time correlations, Reynolds effects) Promising compromise, formulations not stabilized Importance of the Reynolds number should not be neglected!! 134 Airbus - EEA Faculty - Computational Aeroacoustics

135 Concluding remarks Open topics in LES & Direct Noise Computation Interaction of turbulence with shock-waves (BBSAN) Time dependent inflow conditions Impedance in time domain (with a time dependent BL) TBL noise 135 Airbus - EEA Faculty - Computational Aeroacoustics

136 Concluding remarks Strategies in Computational AeroAcoustics Statistical modelling of the radiated sound field (based on RANS) simple but limited description of physics & prediction capacities Acoustic analogy based on a 3-D source volume Lighthill or based on LEE, APE, LEE-SNGR,... not highly recommended, heavy computations (storage), acoustic modelling, interpretation of source terms,... Direct computation of aerodynamic noise (LES, DES) prohibitive cost in low Mach number approximation Surface integral formulations incompressible/compressible : imposed by the importance of mean flow effects (radiation) or by source compacity (diffraction) WEM : Ffowcs Williams & Hawkings / linearized Euler equations (LEE) LEE + mean flow effects, non-linear LEE 136 Airbus - EEA Faculty - Computational Aeroacoustics

137 Acknowledgments «CAA team» Christophe Bogey & Olivier Marsden with the contributions of Xavier Gloerfelt, Sébastien Barré, Julien Berland, Thomas Emmert, Vincent Fleury, Damien Desvignes, Nicolas de Cacqueray, Benoît André, Cyprien Henry Airbus - EEA Faculty - Computational Aeroacoustics