I Estimated numerical schemes in heat convection by OpenFOAM. In this slide, I used linear, upwind and QUICK schemes.

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