The document discusses numerical schemes for estimating heat convection in computational fluid dynamics, highlighting key challenges such as conservativeness, boundedness, and transportiveness when solving diffusion-convection equations. It emphasizes the importance of carefully monitoring the cell Peclet number to prevent issues like overshoot and undershoot, especially in scenarios with high Prandtl and Schmidt numbers. The recommendation is to utilize stabilized numerical methods for resolving complex problems effectively.

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