Code SAFARI by hkksew3563rd

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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)

								
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