estimation of numerical schemes in heat convection by openfoam

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Estimation of numerical schemes in heat convection by OpenFOAM Osaka Univ. Dept. Takuya Yamamoto

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  • Estimation of numerical schemes

    in heat convection by OpenFOAM

    Osaka Univ.Dept.

    Takuya Yamamoto

  • Traps in solving diusion-convection equation

    1. Conserva+veness 2. Boundedness 3. Transpor+veness

    Reference H. K. Versteeg and M. Malalasekera An introduc+on to computa+onal uid dynamics translated Ver. in Japanese ; ,,,

    rela+ve ra+o of convec+on to diusion non-dimensional cell Peclet number

    Pe = FD =u

    /xx ; cell width ; densityFD

    ; momentum ux (= u); diusion conductance (= /x); diusion coecient

  • Indication of numerical schemes

    Linear scheme QUICK schemeBoundedness Boundedness

    Pe < 2 Pe < 83The other condi+ons

    ReferencesH. K. Versteeg and M. Malalasekera An introduc+on to computa+onal uid dynamics translated Ver, in Japanese ; ,,,

    Genera+on of Undershoot Overshoot

  • Ex5.1 in Ref. Book

    T = 1 T = 0

    x = Lx = 0u [m/s]

    condi+on u [m/s] x [m] L [m] Pe [-]

    1 0.1 0.2 1 0.2

    2 2.5 0.2 1 5

    3 2.5 0.05 1 1.25

    T T0TL T0

    =exp ux /( )1exp uL /( )1

    Analy+cal solu+on

    x ; cell width ; density ; diusion coe.

    =1.0 kg/m3 = 0.1 kg/ms

    (kg/ms)

  • Numerical method

    SolverscalarTransportFoam Numerical schemelinear (spatial)steadyState (time)

    Governing Equa+onddx uT( ) =

    ddx

    dTdx

    !

    "#

    $

    %&

  • Ex5.1 in Ref. BookCondi+on 1 Condi+on 2

    condi+on u [m/s] x [m] L [m] Pe [-]1 0.1 0.2 1 0.2

    2 2.5 0.2 1 5

    3 2.5 0.05 1 1.25

    T = 1 T = 0

    x = Lx = 0 u [m/s]

    over- and under-shoot

    Linear scheme

  • Condi+on 2 Condi+on 3

    condi+on u [m/s] x [m] L [m] Pe [-]1 0.1 0.2 1 0.2

    2 2.5 0.2 1 5

    3 2.5 0.05 1 1.25

    T = 1 T = 0

    x = Lx = 0 u [m/s]

    over- and under-shoot

    Linear scheme

    Ex5.1 in Ref. Book

  • Numerical method

    SolverscalarTransportFoam Numerical schemeQUICK (spatial)steadyState (time)

    Governing Equa+onddx uT( ) =

    ddx

    dTdx

    !

    "#

    $

    %&

  • Condi+on 1 Condi+on 2

    condi+on u [m/s] x [m] L [m] Pe [-]1 0.1 0.2 1 0.2

    2 2.5 0.2 1 5

    3 2.5 0.05 1 1.25

    T = 1 T = 0

    x = Lx = 0 u [m/s]

    over- and under-shoot

    QUICK scheme

    Ex5.4 in Ref. Book

  • Condi+on 2 Condi+on 3

    condi+on u [m/s] x [m] L [m] Pe [-]1 0.1 0.2 1 0.2

    2 2.5 0.2 1 5

    3 2.5 0.05 1 1.25

    T = 1 T = 0

    x = Lx = 0 u [m/s]

    over- and under-shoot

    QUICK scheme

    Ex5.4 in Ref. Book

  • Summary

    Be carful for local cell Pe number when we solve diusion-advection equation.

    Be careful especially in high Pr and Sc number, because cell Pe number becomes large.

    We should use stabilized numerical schemes to solve dicult problems.

    Ex) molten metal air water

    Pr O 0.01( )Pr O 1( )Pr 7

  • References

    H. K. Versteeg and M. Malalasekera, An introduction to computational uid dynamics

    translated Ver. in Japanese, ; ,,,