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Introduction Numerical techniques Turbulence Shocks Code SAFARI P. Lafon1 , F. Crouzet2 , F. Daude1 1 LaMSID - UMR EDF/CNRS 2832 2 EDF R&D, AMA Introduction Numerical techniques Turbulence Shocks e Plan de la pr´sentation 1 Basic Numerical Techniques 2 Application to complex geometries 3 Extension to moving grids Introduction Numerical techniques Turbulence Shocks 1 Introduction 2 Numerical techniques Principles DRP schemes Filters 3 Turbulence Physics Modelling 4 Shocks Physics of shocks Modelling Introduction Numerical techniques Turbulence Shocks 1 Introduction 2 Numerical techniques Principles DRP schemes Filters 3 Turbulence Physics Modelling 4 Shocks Physics of shocks Modelling Introduction Numerical techniques Turbulence Shocks Aeroacoustics (1/2) Flow acoustics generated by turbulence Seiner 1984 Flow acoustics interactions with feedback: cavity noise, tones Introduction Numerical techniques Turbulence Shocks Aeroacoustics (2/2) : strategies for CAA For 15 years, a new domain has been developing : Computational AeroAcoustics (CAA) When there is no acoustic feedback on the ﬂow : hybrid methods (two-step method) ﬂow calculation step + acoustic propagation step: incompressible ﬂow + acoustics analogy incompressible ﬂow + linear Euler equations When there is acoustic feedback on the ﬂow Solve the unsteady compressible Navier-Stokes equations to determine the ﬂow ﬁeld and the acoustic ﬁeld in the same computation Towards the Direct Noise Computation Introduction Numerical techniques Turbulence Shocks Aeroacoustics at EDF (1/2) Aeroacoustic phenomena in bypass turbine ducts Flow acoustics coupling in regulation valve Flow acoustics coupling in gate valve Flow acoustics phenomena in safety valve Rarefaction wave in ducts Outdoors acoustic propagation Close problems aeroelasticity combustion instabilities Introduction Numerical techniques Turbulence Shocks Aeroacoustics at EDF (2/2) A code (EOLE) solving linear Euler equations on cartesian grid: acoustic propagation in complex ﬂows A 2D version since 1987 A 3D version since 1995 A code (EULER2D) solving non linear Euler equations on 2D curvilinear grids: aeroacoustic coupling Several applications to ﬂow acoustics industrial problems Aeroelasticity A collaboration with Ecole Centrale Lyon : Christophe Bailly, Christophe Bogey, Olivier Marsden High order optimized ﬁnite diﬀerence schemes Extension to curvilinear grids Several thesis especially Thomas Emmert (2007) Introduction Numerical techniques Turbulence Shocks SAFARI e SAFARI : Simulation A´roacoustique dans les Fluides Avec e R´sonances et Interactions Main objective : SAFARI = EOLE + EULER2D + ECL High order discretization on structured grids Multi-domain for complex geometries Euler, Navier-Stokes and linear Euler equations Moving grids for aeroelasticity Introduction Numerical techniques Turbulence Shocks Fluid models Euler equations ∂ρ ∂ + (ρuj ) = 0 ∂t ∂xj ∂ρui ∂ + (ρui uj + p δij ) = 0 ∂t ∂xj ∂ρet ∂ + [(ρet + p) uj ] = 0 ∂t ∂xj Introduction Numerical techniques Turbulence Shocks Fluid models Navier-Stokes equations ∂ρ ∂ + (ρuj ) = 0 ∂t ∂xj ∂ρui ∂ ∂σij + (ρui uj + p δij ) = ∂t ∂xj ∂xj ∂ρet ∂ ∂σjk uk ∂qj + [(ρet + p) uj ] = − ∂t ∂xj ∂xk ∂xj Introduction Numerical techniques Turbulence Shocks Fluid models Linear Euler equations ∂ρ ∂ + ρ u0j + ρ0 uj = 0 ∂t ∂xj ∂ρui ∂ ∂u0i + ρui u0j + p δij + ρuj + ρ u0j = 0 ∂t ∂xj ∂xj ∂p ∂ ∂uj + (puj ) + (γ − 1) p = 0 ∂t ∂xj ∂xj Introduction Numerical techniques Turbulence Shocks 1 Introduction 2 Numerical techniques Principles DRP schemes Filters 3 Turbulence Physics Modelling 4 Shocks Physics of shocks Modelling Introduction Numerical techniques Turbulence Shocks Principles Need of high order for aeroacoustics A simple one-dimensional wave equation Tam 1995 (a) fourth-order central diﬀerence scheme (b) sixth-order central diﬀerence scheme (c) DRP scheme (seven-point stencil) Introduction Numerical techniques Turbulence Shocks Principles High order numerical techniques Mac-Cormack Discontinuous Galerkin High order ﬁnite volume High order ﬁnite diﬀerence → Dispersion Relation Preserving (DRP) schemes (Tam 1995, Bogey & Bailly 2004) Introduction Numerical techniques Turbulence Shocks DRP schemes High order spatial derivatives A ﬁnite diﬀerence scheme can be transformed in Fourier space N N ∂f 1 (x0 ) = aj f (x0 + j∆x) → ks ∆x = 2 aj sin(jk∆x) ∂x ∆x j=−N j=1 For ks to be real (no dissipation), aj must be antisymmetric → Centered Finite Diﬀerence Schemes Optimized schemes can be obtained by minimizing the dispersion error ln(k∆x)h |ks ∆x − k∆x| d(ln(k∆x)) ln(k∆x)l → A set of optimized coeﬃcients aj Introduction Numerical techniques Turbulence Shocks DRP schemes Optimized schemes: Dispersion Relation Preserving Eﬀective wave number and associated error Introduction Numerical techniques Turbulence Shocks Filters Need of a ﬁltering procedure Non dissipative centered ﬁnite-diﬀerence schemes do not damp spurious short waves due to numerical instabilities or non linearities A ﬁltering procedure has to be apply on the solution N u f (x0 ) = u (x0 ) − σd Gu (x0 ) Gu (x0 ) = dj u(x0 + j∆x) j=−N N Fourier transform → Gk (k∆x) = d0 + 2dj cos(jk∆x) j=1 For having no dispersion, dj must be symmetric Optimized ﬁlters can be obtained by selecting only the spurious short waves to remove Introduction Numerical techniques Turbulence Shocks Filters Optimized ﬁlters Eﬀective wave number and associated error Introduction Numerical techniques Turbulence Shocks 1 Introduction 2 Numerical techniques Principles DRP schemes Filters 3 Turbulence Physics Modelling 4 Shocks Physics of shocks Modelling Introduction Numerical techniques Turbulence Shocks Physics Large scales and small scales Turbulence is a multi-scale phenomena Brown & Roshko JFM 1974 From large energetic scales to small dissipative scales Anderson 2004 Introduction Numerical techniques Turbulence Shocks Physics Modelling strategies for turbulence All scales are computed : Direct Numerical Simulation (DNS) expensive because very ﬁne grids are needed Only scales captured by the grid are computed and the smallest ones are modelled : Large Eddy Simulation (LES) Applicable to industrial conﬁgurations All scales are modelled : Reynolds Averaged Navier-Stokes Industrial applications Introduction Numerical techniques Turbulence Shocks Modelling RANS methods Reynolds Average Navier Stokes methods Navier Stokes equations are a priori averaged: ¯ 1- Reynolds decomposition ui = ui + u i 2- Averaging of the equations u ∂¯i u¯ ∂¯i uj ∂¯ p ∂ui uj ¯ ∂ σij ρ + + = −ρ + + fi ext (1) ∂t ∂xj ∂xi ∂xj ∂xj Reynolds stresses are treated as viscous stresses: t ∂¯i u u ∂¯j 2 σij = −ρ ui uj = µt + − ρkδij ∂xj ∂xi 3 with the turbulent eddy viscosity model µt = ρCµ (k 2 / ) Introduction Numerical techniques Turbulence Shocks Modelling LES methods (1/2) From resolved scales to not resolved scales Anderson 2004 Is it possible to take into account the small non resolved dissipative scales without disturbing the resolved scales? Introduction Numerical techniques Turbulence Shocks Modelling LES methods (2/2) : two main approaches The dissipative scales are taken into account through an eddy viscosity model linked to mesh size 1 ∂˜i u u ∂˜j µt = (Cs ∆)2 s s 2˜ij ˜ij u s o` ˜ij = + 2 ∂xj ∂xi The classical approach well established in the literature Sophisticated models are needed to control the turbulent viscosity The ﬁltering needed to take into account the dissipative scales is made by the numerical selective ﬁlters Add no computational cost Add no viscosity, so adapted to aeroacoustic computations Introduction Numerical techniques Turbulence Shocks 1 Introduction 2 Numerical techniques Principles DRP schemes Filters 3 Turbulence Physics Modelling 4 Shocks Physics of shocks Modelling Introduction Numerical techniques Turbulence Shocks Physics of shocks Discontinuities in compressible ﬂows Shocks are very thin discontinuities appearing in compressible ﬂows Chassaing 2000 Upstream and downstream the shock, ﬂow behavior is non dissipative but through the shock, viscous eﬀects induces entropy variation No grid can capture the shock: special modelling is needed Introduction Numerical techniques Turbulence Shocks Physics of shocks Theoretical concepts Weak solutions : smooth solutions + discontinuities Conservative form of the equations f (u) ∂u ∂t + ∂x =0 → Uin+1 = Uin − ∆t ∆x Fi+ 1 − Fi− 1 2 2 Fi+ 1 = n , Un n F (Ui−p i−p+1 , ..., Ui+q ) 2 Shock speed and jump conditions are well captured Monotonicity No new local extrema may be created Shock capturing schemes have been developed based on the hyperbolic theory of conservation laws with monotone and high resolution properties at shocks (ex: TVD schemes, EULER2D) Introduction Numerical techniques Turbulence Shocks Physics of shocks Application to the shock tube problem LeVeque 1992 Introduction Numerical techniques Turbulence Shocks Modelling Shock-capturing procedure (1/2) Non-linear shock-capturing ﬁlter (Kim & Lee AIAA J. 2001): n+1 ∗ x Ui,j,k = Ui,j,k + ∆t Di,j,k with the dissipation term x 1 Di,j,k = Di+1/2 − Di−1/2 ∆x where the shock capturing numerical dissipation ﬂux vector writes (2) ∗ ∗ Di+1/2 = ∆|λ|x i+1/2 i+1/2 Ui+1,j,k − Ui,j,k Introduction Numerical techniques Turbulence Shocks Modelling Shock-capturing procedure (2/2) the stencil eigenvalue 3 3 ∆|λ|x x x i+1/2 = max (|λ |i+m,j,k ) − min (|λ |i+m,j,k ) m=−2 m=−2 the non-linear dissipation function (2) 3 x i+1/2 = κj,k max (νi+m,j,k ) m=−2 with the Jameson detector x |pi−1,j,k − 2pi,j,k + pi+1,j,k | νi,j,k = pi−1,j,k + 2pi,j,k + pi+1,j,k and κj,k being an adaptative control constant. Introduction Numerical techniques Turbulence Shocks Modelling Conservativity ∂u f (u) The general conservation equation ∂t + ∂x = 0 shall be discretized Finite Diﬀerence Finite Volume j=N ∆t ∆t Uin+1 = Uin − aj fi+j Uin+1 = Uin − F 1 − Fi− 1 ∆x ∆x i+ 2 2 j=−N j=N Fi+ 1 = j=1 bj (fi+j − fi+j−1 ) 2 For a three point stencil, the equivalence between the two formulations writes: 1 bj = aj = 2 And for a general centered stencil, one can show that: N bj = al l=j Introduction Numerical techniques Turbulence Shocks Modelling Examples of schock tube computations with SAFARI Sod Tube Case Lax Tube Case Introduction Numerical techniques Turbulence Shocks Modelling Consequences on the treatment of small scales Truc a a von K´rm´n-Saﬀman spectrum E (k) k g grid cut-oﬀ wavenumber (2 pts / λ) k f ﬁltering cut-oﬀ wavenumber (4 pts / λ) Introduction Numerical techniques Turbulence Shocks Modelling Consequences on the treatment of small scales Truc a a von K´rm´n-Saﬀman spectrum E (k) E (k) with eddy viscosity model k g grid cut-oﬀ wavenumber (2 pts / λ) k f ﬁltering cut-oﬀ wavenumber (4 pts / λ) Introduction Numerical techniques Turbulence Shocks Modelling Consequences on the treatment of small scales Truc a a von K´rm´n-Saﬀman spectrum E (k) E (k) with eddy viscosity model E (k) with explicit selective ﬁltering k g grid cut-oﬀ wavenumber (2 pts / λ) k f ﬁltering cut-oﬀ wavenumber (4 pts / λ)