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  • Simulation numriquedes coulements complexes

    coulement moyen et viscosit sous-maille

    en collaboration avec

    - Pierre Borgnat , Hatem Touil (LP, ENSL)- Adrien Cahuzac, Jrme Boudet, Jolle Caro

    Liang Shao, Jean-Pierre Bertoglio (LMFA, ECL)- Guillaume Balarac (LEGI, Grenoble)- Federico Toschi (TU Eindhoven)

  • Inhomognit - phnomnologie

    !

    " = #u2

    : cisaillement

    turbulence homogne et isotropeturbulence cisaille

    coulement moyen + fluctuations

    tourbillon

    l 3/23/1)(/ lll !"" #$uteddy!

    "1

    sheart

    l

    !"!'u

    L

    Kolmogorovs theory

    !"#2'u$

    taux de dissipation dnergie

    !

    u = u + " u : dcomposition de Reynolds

    coulement autour dun cylindre ReD=47000 (rgime turbulent sous-critique)

  • Simulation numrique descoulements turbulents

    goal= get a physically-sound numerical representation of the flow

    -Direct Numerical Simulation (all scales of motion) is usually prohibitive

    - alternatives = partial representation of the flow : - Reynolds-Averaged Navier-Stokes : mean flow

    - Large-Eddy Simulation : large-scale dynamics of the flow

    Does it make sense (from a physical viewpoint) ? What are the governing equations?

    subgridgridgridt UU !" div)~(NS

    ~+=

    Ssubgridsubgridsubgrid~

    2I)(tr !"" #=#

    SGS viscosity = property of the flow, not of the fluid (very) bad news !

    ?)etc.,~

    ,~,( gridgridsubgrid UU !"#

  • 4

    shearuniformlocally:j

    j

    i

    iix

    x

    uuu !+"=

    #

    #

    i

    22

    2dS),(),(),(

    4

    1)(T ! ""

    #

    $

    %%

    &

    '()+()(=

    lB

    j

    j

    iisgs lxul

    x

    ulxulxu

    ll

    *

    +*

    *++

    ,

    !

    " # sgs $ S = 2 % sgs(&) $ S 2

    ' Tsgs(&) & with & = grid spacing

    - after some (exact) calculations from the Navier-Stokes equations

    !

    " # u ($) % # S & $ and "U($) % S & $ ' ( sgs($) = (Cs$)2& S ) S ( )

    - in the context of LES :

    Effet du cisaillement en quations

    Lvque et al., J. Fluid Mech. (2007)Shear-Improved Smagorinsky Model

  • 5

    If : Smagorinsky model for HI turbulence (Cs =0.17)

    If : Shear-dominated SGS viscosity

    SS~~

    >>!

    SS~~!>>

    ( ) ( ) SCu

    C sssgs~)( 22!"#$

    #

    #!"#$%

    &

    ( )S

    SCssgs ~

    ~ 2

    2!

    "#$%

    MINUS :PLUS : physically-sound, no ad-hoc parameter low computational cost easy to parallelize convenient for boundary conditions

    requires estimation of mean-flowas the simulation runs!

    !

    " sgs(#) = (Cs#)2$ S % S ( )

    signal processing

  • Application to complex-geometryunsteady turbulent flows (real-world flows)

    How to extract (or reconstruct) the mean flow as the simulation runs ?

    - physically sound : smoothing in time mean flow = low-frequency component of the flow

    -manageable from computational viewpoint : recursive filter-adaptive : Kalman filtering

    Cahuzac et al., in press Phys. Fluids (2010)

  • grid cut-offsmoothingcut-off (fc)

    SGS

    very-largescales resolved

    scales

    SISM

    f

    Euu(f)in

    stab

    ilitie

    s

    turb

    ulen

    ce cascade

  • Exponentially-weighted moving averageexponential smoothing

    !

    cexp =2" fc #t

    3$ 3.628 fc #t

    conceptually simple local in space low memory storage

    MINUS :PLUS :

    uniform physically-sound cut-off frequency ? unavoidable phase delay :

    !

    "t / cexp

    )1(

    exp

    )(

    exp

    )1()1( +

    +!+!"=

    nnn

    ucucu

    -cexp : smoothing factor- equivalent tp first-order low pass filter with cut-off frequency fc :

    at each grid point

  • Low-pass Kalman filter adaptive exponential smoothing

    )1()()1( +++=

    nnn

    uuu !

    control increment = random noise with fixed variance

    !

    "# u

    =2$ fc %t

    3& u

    *

    )1()1()1( ++++=

    nnnuuu !

    deviation from the estimated mean = random noise with variance

    : observation

    : model

    K(n+1) optimal Kalman gain (given by theory)!

    "#u(n+1)

    =max u* $ u(n+1)

    % u(n+1)

    , 0.1$ u*2[ ]

    periodsstationaryfor/ .expcuu =!! ""

    )1()1()()1()1( )1( ++++

    !+!"=nnnnn

    uKuKu

    why ?

    predict and update :

  • 10!/*ufc =

    !

    u*

    = uwall

    3.106 mesh-pointsr+=1, R+=20, z+=25

    resolution: 49x89x41x+=52, z+=31, y+=0.524

    !

    St = fs "D /U#

    !

    fc = 2 f s

    Strouhal numberfriction velocity

    Practical evaluation of the methodTurbFlow solver (Jrme Boudet, LMFA):

    - Finite volume- Convective fluxes by centered schemes on four points- Diffusive fluxes by centered scheme on two points- SGS model = Shear-Improved Smagorinsky Model (SISM)

    plane-channel Flow at Rew = 395 flow past a circular cylinder at ReD = 47,000

    Cahuzac et al. Phys. Fluids (2010)

  • 11

    spatial average (along x-z)

    exponential smoothing compared tospatial average (along x)

    Kalman filtering

    Velocity probes at y+=10 in the plane-channel flow (Rew=395)

  • 12

    streamwise velocity profile turbulent intensity profiles

    Plane-channel flow

    comparisons with DNS data

    friction velocity:

    1.59.0!

    = smuDNS

    w

    1.exp.54.0!

    = smuw

    within the 10% level of accuracyreviewed for standard SGS models

  • 13

    spanwise spectraof streamwise velocity

    plane-channel flow

    comparisons with DNS data

    two-point velocity correlationfor spanwise separation

  • 14

    plane-channel flow

    comparisons with DNS data

    two-point correlation functions promising for aero-acoustics

    streamwise

    spanwise

  • !

    Cp =p " p#1

    2$U#

    2

    !

    Cf ="w

    1

    2#U$

    2

    % Re

    2

    2

    1

    !

    ="

    U

    pC rmsp

    #

    flow past a circular cylinder

    Friction and pressure coefficients

  • flow past a circular cylinder

    mean flow andturbulence intensities

    Work in progress...

  • flow past a circular cylinder

    comparisons with experimental dataspectra in frequency

    (a)

    (a)(b)

    (b)

    (c)

    (c)

    1.8 D

    promising for aero-acoustics

    fc

  • Adaptivity of the Kalman filter

  • flow past a circular cylinder

    a drawback of the smoothing: phase delay

    KalmanKt/!

    10

  • Conclusion

    from a physical viewpoint :- take into account mean-flow inhomogeneity + unsteadiness :

    SISM (no ad-hoc parameter) + (adaptive) temporal smoothing

    from a numerical viewpoint :- inconditionnally stable- low cost algorithms- easy to parallelize + convenient for boundary conditions

    To-Do list:- address realistic turbo-machinery flow / aero-acoustic features

    Flocon project (LMFA) : Adrien Cahuzac + Jrme Boudet- reduce phase delay :

    - improve Kalman filter- consider Lagrangian filtering (along fluid trajectory)

  • Plane channel flow (academic flow)Lvque et al., J. Fluid Mech. (2007)Boudet et al., J. of Therm. Science. (2008)Cahuzac et al. Phys. Fluids (2010)

  • 22

    flow past a circular cylinder

    comparisons with experimental data

    flow characteristics

  • Mesh resolution

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