![Ex5.1 in Ref. Book
x
=
0 x
=
L
T
=
1
T
=
0
u
[m/s]
condi+on
u
[m/s]
δx
[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
Analy+cal
solu+on
T −T0
TL −T0
=
exp(ρux /Γ )−1
exp(ρuL /Γ )−1
δ x ;
ρ =1.0 kg/m3
Γ = 0.1 kg/m・s
cell
width
ρ ;
density
Γ ;
diffusion
coeff.
(kg/m・s)](https://image.slidesharecdn.com/peeng-141214061635-conversion-gate02/85/Estimation-of-numerical-schemes-in-heat-convection-by-OpenFOAM-4-320.jpg)
![Ex5.1 in Ref. Book
T
=
1
Condi+on
1
Condi+on
2
condi+on
u
[m/s]
δx
[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
=
0
x
=
0
u
[m/s] x
=
L
over-‐
and
under-‐shoot
Linear
scheme](https://image.slidesharecdn.com/peeng-141214061635-conversion-gate02/85/Estimation-of-numerical-schemes-in-heat-convection-by-OpenFOAM-6-320.jpg)
![T
=
1
Ex5.1 in Ref. Book
Condi+on
2
Condi+on
3
condi+on
u
[m/s]
δx
[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
=
0
x
=
0
u
[m/s] x
=
L
over-‐
and
under-‐shoot
Linear
scheme](https://image.slidesharecdn.com/peeng-141214061635-conversion-gate02/85/Estimation-of-numerical-schemes-in-heat-convection-by-OpenFOAM-7-320.jpg)
![T
=
1
Ex5.4 in Ref. Book
Condi+on
1
Condi+on
2
condi+on
u
[m/s]
δx
[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
=
0
x
=
0
u
[m/s] x
=
L
over-‐
and
under-‐shoot
QUICK
scheme](https://image.slidesharecdn.com/peeng-141214061635-conversion-gate02/85/Estimation-of-numerical-schemes-in-heat-convection-by-OpenFOAM-9-320.jpg)
![T
=
1
Ex5.4 in Ref. Book
Condi+on
2
Condi+on
3
condi+on
u
[m/s]
δx
[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
=
0
x
=
0
u
[m/s] x
=
L
over-‐
and
under-‐shoot
QUICK
scheme](https://image.slidesharecdn.com/peeng-141214061635-conversion-gate02/85/Estimation-of-numerical-schemes-in-heat-convection-by-OpenFOAM-10-320.jpg)
Estimation of numerical schemes in heat convection by OpenFOAM
- 1. Estimation of numerical schemes in heat convection by OpenFOAM Osaka Univ. Dept. Takuya Yamamoto
- 2. Traps in solving diffusion-‐‑‒convection equation 1. Conserva+veness 2. Boundedness 3. Transpor+veness Reference H. K. Versteeg and M. Malalasekera An introduc+on to computa+onal fluid dynamics translated Ver. in Japanese 訳; 松下洋介,斎藤泰洋,青木秀之,三浦隆利 数値流体力学 森北出版 rela+ve ra+o of convec+on to diffusion non-‐dimensional cell Peclet number Pe = F D = ρu Γ /δ x δ x ; cell width ρ ; density F D Γ ; momentum flux (= ρu) ; diffusion conductance (= Γ/δx) ; diffusion coefficient
- 3. Indication of numerical schemes • Linear scheme • QUICK scheme Boundedness Boundedness Pe 2 Pe 8 3 The other condi+ons References H. K. Versteeg and M. Malalasekera An introduc+on to computa+onal fluid dynamics translated Ver, in Japanese 訳; 松下洋介,斎藤泰洋,青木秀之,三浦隆利 数値流体力学 森北出版 Genera+on of • Undershoot • Overshoot
- 4. Ex5.1 in Ref. Book x = 0 x = L T = 1 T = 0 u [m/s] condi+on u [m/s] δx [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 Analy+cal solu+on T −T0 TL −T0 = exp(ρux /Γ )−1 exp(ρuL /Γ )−1 δ x ; ρ =1.0 kg/m3 Γ = 0.1 kg/m・s cell width ρ ; density Γ ; diffusion coeff. (kg/m・s)
- 5. Numerical method • Solver scalarTransportFoam • Numerical scheme linear (spatial) steadyState (time) • Governing Equa+on d dx (ρuT) = d dx Γ dT dx ! # $ %
- 6. Ex5.1 in Ref. Book T = 1 Condi+on 1 Condi+on 2 condi+on u [m/s] δx [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 = 0 x = 0 u [m/s] x = L over-‐ and under-‐shoot Linear scheme
- 7. T = 1 Ex5.1 in Ref. Book Condi+on 2 Condi+on 3 condi+on u [m/s] δx [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 = 0 x = 0 u [m/s] x = L over-‐ and under-‐shoot Linear scheme
- 8. Numerical method • Solver scalarTransportFoam • Numerical scheme QUICK (spatial) steadyState (time) • Governing Equa+on d dx (ρuT) = d dx Γ dT dx ! # $ %
- 9. T = 1 Ex5.4 in Ref. Book Condi+on 1 Condi+on 2 condi+on u [m/s] δx [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 = 0 x = 0 u [m/s] x = L over-‐ and under-‐shoot QUICK scheme
- 10. T = 1 Ex5.4 in Ref. Book Condi+on 2 Condi+on 3 condi+on u [m/s] δx [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 = 0 x = 0 u [m/s] x = L over-‐ and under-‐shoot QUICK scheme
- 11. Summary • Be carful for local cell Pe number when we solve diffusion-‐‑‒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 difficult problems. Ex) molten metal air water Pr ≈ O(0.01) Pr ≈ O(1) Pr ≈ 7
- 12. References • H. K. Versteeg and M. Malalasekera, “An introduction to computational fluid dynamics” translated Ver. in Japanese 数値流流体⼒力力学, 訳; 松下洋介,斎藤泰洋,⻘青⽊木秀 之,三浦隆利利 森北北出版