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1 Turbulence et Génération de Bruit Equipe de recherche du Centre Acoustique LMFA, UMR CNRS 5509, Ecole Centrale de Lyon Simulation Numérique en Aéroacoustique Institut Henri Poincaré - 16 novembre 2006 ËÑ ÖØ ØÓÒ ÓÔØÑ Ò Ð³ Ô ÓÙÖÖ C. Bogey, J. Berland & C. Bailly LMFA, UMR CNRS 5509 Ecole Centrale de Lyon

2 ÇÙØÐÒ ¹ ÊÓ ÑÔ Background / Motivations Spatial discretization optimized in the Fourier space Finite differences for spatial derivatives Selective filters for removing high-frequency waves Time integration : optimized Runge-Kutta schemes Applications Acoustic test problem : diffraction by a cylinder Navier-Stokes simulations (Large-Eddy Simulations) Concluding remarks ¾

3 Development of Computational AeroAcoustics (CAA) vorticity field of a coaxial jet ÅÓØÚØÓÒ direct simulation of sound generation by solving the unsteady Navier-Stokes equations for compressible flows sound pressure field simulations of long-range propagation (Linearized Euler Equations) Key numerical issues in CAA disparities in magnitudes and length scales between flow and acoustics turbulent and sound broadband spectra far-field propagation

4 Ù Ù Ü ¼ Ø ÅÓØÚØÓÒ Problem model for wave propagation 1-D advection equation : ¼µ ܵ Ù Ü exact solution Ù Ü Øµ Ü Øµ by Fourier-Laplace transform : dispersion relation elementary solution : harmonic plane wave Ü Øµ numerical approximation : Development of schemes optimized in the Fourier space space : Dispersion-Relation-Preserving schemes of Tam & Webb (JCP, 1991), spectral-like schemes of Lele (JCP, 1992) time : Runge-Kutta algorithms of Hu et al. (JCP, 1996)

5 ÇÙØÐÒ ¹ ÊÓ ÑÔ Background / Motivations Spatial discretization optimized in the Fourier space Finite differences for spatial derivatives Selective filters for removing high-frequency waves Time integration : optimized Runge-Kutta schemes Applications Acoustic test problem : diffraction by a cylinder Navier-Stokes simulations (Large-Eddy Simulations) Concluding remarks

6 Ù ³ Ù Ü Ò Ò Ù Ð Ù ³ Ü Ü ½ Ò Ü ÒØ «ÖÒ ÓÖ ÔØÐ ÖÚØÚ Explicit finite-difference schemes 1 Ù ½ ³ Ü Ð Ü Ð Ò Ü Ð Ñ Ü Ü Ð Ð Ü centered Ñ Ò upwind Ü Ñ Ð Ñ Ò Particular case of the continuous relation: ½ Ò Ù Ü Üµ By Fourier transform: Ü Ñ Ù Üµ ½ Ù µ Ù µ Ü ½ Ñ Numerical dimensionless wavenumber Ü Ñ

7 Ò ¾ Centered schemes (n = m) is antisymmetric π 3π/4 ÒØ «ÖÒ ÓÖ ÔØÐ ÖÚØÚ is real (no dissipation) Ü π/2 π/4 Ü ¾ Ò Ü 0 0 π/4 π/2 3π/4 π ½ high-order schemes : are determined by cancelling the Taylor series formal truncation Ç Ü order µ ¾Ò low-dispersion schemes : are determined by minimizing the error between the exact and numerical wavenumbers and over a large Ü wavenumber range Ð Ü Ü Ù Ü e.g. in Bogey & Bailly (JCP, 2004), minimization of the integral error Ü Ü ÐÒ Üµ Ü ½

8 ¾ ÒØ «ÖÒ ÓÖ ÔØÐ ÖÚØÚ Non-centered schemes (n m) has an imaginary part (providing dissipation/amplification) approximate solution for the 1-D advection equation Ù Ü Øµ Ê µ Ø ßÞ Ð phase error ÁÑ µø ßÞ Ð dissipation /amplification Ü Øµ ßÞ Ð exact solution minimization both of phase error and of dissipation/amplification e.g. in Berland et al. (JCP, 2006) with the integral error ½ «µ Ü Ê Üµ ßÞ Ð ½ dispersion ßÞ Ð dissipation ܵ Ü «ÁÑ Üµ low-dispersion and low-dissipation schemes

9 ÒØ «ÖÒ ÓÖ ÔØÐ ÖÚØÚ Numerical wavenumber k s x vs. exact wavenumber k x for centered schemes π 3π/4 2nd, 4th, 6th, 8th, 10th and 12th-order central differences DRP 7-pt Tam & Webb (1993) DRP 15-pt Tam (2003) k s x π/2 π/4 11-pt and 13-pt schemes of Bogey & Bailly (2004) tridiag. 5-pt 6th-order and compact 8th-order schemes, pentadiag. 4th-order of Lele (1992) 0 0 π/4 π/2 3π/4 π k x prefactored 5-pt 4th-order Æ scheme of Ashcroft & Zhang (2003)

10 ÒØ «ÖÒ ÓÖ ÔØÐ ÖÚØÚ Phase-velocity error in terms of points per wavelength 10 0 k x k s x /π error observed for a harmonic wave Ü Øµ propagating at Ú ³ accuracy limit given by 10 6 Ü ½¼ Ú ³ Ü λ/ x ½¼

11 Ú ½ ½¼ ÒØ «ÖÒ ÓÖ ÔØÐ ÖÚØÚ Group-velocity error as a function of the exact wavenumber propagation of a wave-packet at the group velocity Ú µ dk s /dk dk s /dk π/4 π/2 3π/4 π k x π/4 π/2 3π/4 π k x accuracy limit given by ½½

12 «ÖÒ ÓÖ ÔØÐ ÖÚØÚ ÒØ Accuracy limits scheme E vϕ E vg k x max λ/ x min k x max λ/ x min k max x k vϕ /k max CFD 2nd-order CFD 4th-order CFD 6th-order CFD 8th-order CFD 10th-order CFD 12th-order DRP 7-pts 4th-order DRP 15-pts 4th-order OFD 11-pts 4th-order OFD 13-pts 4th-order CoFD 6th-order CoFD 8th-order CoFD opt. 4th-order Opt. pre. 4th-order ½¾

13 Ò Need for spatial filtering ËÐØÚ ÔØÐ ÐØÖÒ grid-to-grid oscillations are not resolved by F-D schemes according to the Nyquist-Shannon theorem the highest wave-numbers, poorly resolved by F-D, must be removed without affecting the long (physical) waves accurately discretized k x = π/4 k x = π/2 k x = π λ/ x = 8 λ/ x = 4 λ/ x = 2 Explicit discrete filtering (Æ Æ Æ Æ) Ù Ü µ Ù Ü µ Ù Ü Üµ Ñ transfer function in the Fourier space Ò Ü Üµ ½ Ñ ½

14 ÔØÐ ÐØÖÒ ËÐØÚ Requirements on the transfer function stability : ܵ ½ removal of grid-to-grid oscillations : µ ¼ normalization : ¼µ ½ Centered filters Non-centered filters is real (no dispersion) (see fig.) has an imaginary part (phase error) Optimized filtering by choosing minimizing an integral error ½ π/4 π/2 3π/4 π Ü e.g., ¾ ½ «µ½ ܵ ßÞ Ð ½ dissipation ܵ ܵ ßÞ Ð «Ü phase error ½

15 Transfer function ½ of centered filters ËÐØÚ ÔØÐ ÐØÖÒ 1 G k nd, 4th, 6th, 8th, 10th and 12th-order standard explicit filters optimized 7-pt 2nd-order filter of Tam & Webb (1993) optimized 11-pt 2nd-order and 13-pt 4th-order filters of Bogey & Bailly (2004) λ/ x implicit tridiag. 2nd, 4th, 6th, 8th and 10th implicit filters ¼µ see Lele (1992), «Gaitonde & Visbal (2000) ½

16 ÇÙØÐÒ ¹ ÊÓ ÑÔ Background / Motivations Spatial discretization optimized in the Fourier space Finite differences for spatial derivatives Selective filters for removing high-frequency waves Time integration : optimized Runge-Kutta schemes Applications Acoustic test problem : diffraction by a cylinder Navier-Stokes simulations (Large-Eddy Simulations) Concluding remarks ½

17 Ù Ò ½ ¼ Ò Ù ÌÑ ÒØÖØÓÒ Explicit Runge-Kutta schemes differential equation Ù Ò Ø ÙÒ Øµ Ù Ò Üµ Ù Ü Ò Øµ General form of a low-storage Runge-Kutta scheme at Ô-stages : Ô ½ ½ Ù Ò ½ Ù Ò Ø Ã Ø Ò Ø Ã with à ½ ½ By Fourier analysis, numerical amplification factor Ê Ê ÙÒ ½ Ô Øµ exact factor : Ê Ø ½ Optimized Runge-Kutta schemes the coefficients are determined by minimizing the errors over a large range of pulsations Ø ½

18 Damping factor (dissipation) ÌÑ ÒØÖØÓÒ 1 R s (ω t) π/4 π/2 3π/4 π 5π/4 ω t 4th-order Prince (1980) 8th-order of Dormand & optimized 2nd-order Ô 2N of Bogey & Bailly (2004) and µ 4th-order Ô µ 2N of Berland et al. (2006) 4th-order 2N Ô µ of Car- penter & Kennedy (1994) LDDRK46 and LDDRK56 of Hu et al. (1996) opt. 4th-order 2N Ô µ of Stanescu & Habashi ½

19 Phase error (dispersion) 10 0 ÌÑ ÒØÖØÓÒ 10 1 ω t ω s t /π Ê Ê Ê Ø Øµ π/4 π/2 3π/4 π ω t ½

20 ÌÑ ÒØÖØÓÒ Accuracy limits dissipation ½ Ê Øµ & dispersion ³ Ø Ø CFL number given for the opt. 11-pt FD scheme scheme formal E d E ϕ stability order ω t max β ω t max β ω t max β Standard RK4 4th Standard RK8 Dormand et al. 8th Stanescu et al. 4th Carpenter & Kennedy 4th Opt. LDDRK46 Hu et al. 4th Opt. LDDRK56 Hu et al. 4th Opt. 2N-RK Bogey et al. 2nd Opt. 2N-RK Berland et al. 4th ¾¼

21 ÇÙØÐÒ ¹ ÊÓ ÑÔ Background / Motivations Spatial discretization optimized in the Fourier space Finite differences for spatial derivatives Selective filters for removing high-frequency waves Time integration : optimized Runge-Kutta schemes Applications Acoustic test problem : diffraction by a cylinder Navier-Stokes simulations (Large-Eddy Simulations) Concluding remarks ¾½

22 Ì Ø ÔÖÓÐÑ ¹ ÓÙ Ø «ÖØÓÒ ¾¹ Acoustic diffraction by a cylinder (2nd CAA Workshop, 1997) non-compact monopolar source scattering by the cylinder complex diffraction pattern sensitive to numerical accuracy Numerical methods : optimized 11-pt F-D & filters and 6-stage Runge-Kutta 2 configurations of algorithms near the wall centered F-D & filters non-centered optimized 11-point F-D & filters ¾¾

23 Results ¾¹ Ì Ø ÔÖÓÐÑ ¹ ÓÙ Ø «ÖØÓÒ Directivity µ ÖÔ ¼¾ at Ö x analytical computation x analytical computation 2 2 µ µ centered schemes at the wall non-centered schemes at the wall ¾

24 ÑÙÐØÓÒ ÄÖ¹Ý ËÑÙÐØÓÒ µ ÆÚÖ¹ËØÓ Large Eddy Simulation (LES) the turbulent structures supported by the grid are computed the (dissipative) effects of the subgrid scales are modelized LES based on explicit filtering 1/L k sf c E(k) k grid viscous dissipation c 1/λ 1/η resolved filtered subgrid scales scales scales k energy draining taken into account by the filtering resolved scales calculated accurately by the F-D scheme, unaffected by the filtering nor by the time integration flow features independent of the numerics the use of optimized schemes appears appropriate for LES ¾

25 ÑÙÐØÓÒ ÄÖ¹Ý ËÑÙÐØÓÒ µ ÆÚÖ¹ËØÓ Investigation of noise generation subsonic and supersonic jets cavity noise airfoil noise Noise generated by a subsonic jet (Mach Reynolds 500,000) Velocity contour and pressure field Far-field pressure spectra at ¼ Æ p (db/st) St, simulations measurements ¾

26 ÑÙÐØÓÒ ÄÖ¹Ý ËÑÙÐØÓÒ µ ÆÚÖ¹ËØÓ Investigation of turbulence simulations under controlled (physical and numerical) conditions direct calculation of flow quantities including dissipation Energy budget in a turbulent jet Mach Reynolds 11,000 Self-similarity region for ½¾¼Ö ¼ Ü ½¼Ö ¼ Vorticity norm y/δ 0.5 convection : production : dissipation : turb. diffusion : pres. diffusion : LES, o expe LES, o expe LES, o expe LES, o expe LES ¾

27 ÓÒÐÙÒ ÖÑÖ Optimized finite-difference methods accurate / simple / efficient treatment of boundary conditions non-uniform and curvilinear mesh complex geometries : interpolation, overset grids, multi-domain explicit selective filtering for Large Eddy Simulations (LES) Some difficulties large stencils (multi-domain / parallelization) treatment of shock-waves ¾