Estimation of numerical schemes in heat convection by OpenFOAM Osaka Univ. Dept. Takuya Yamamoto
Jul 13, 2015
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, ; ,,,