未來展望

第五章結論與展望

5.2 未來展望

目前分析上遇到一些現象可供未來繼續研究的方向之整理如下：

（1）探討完全對稱之 0

^o

與 180

^o

傾斜角度下自然對流模擬上無法收斂之原因。

（2）增加網格數或是更高階之速度與溫度梯度微分以處理高雷諾數下之剪切流、高瑞利數之自然對流以及低理查森數之混合對流分析。

（3）探討不同紊流模式之熱傳預測能力。

參考文獻

[1] U. Ghia, K. N. Ghia, C. T. Shin. High-Re solutions for incompressible flow using the Navier-Stokes equations and a Multigrid Method. Journal of computational physics 48 (1982) 387-411.

[2] G. De Vahl Davis. Natural connection of air in a square cavity a bench mark numerical solution. International Journal for Numerical Methods in Fluids, vol. 3 (1983) 249-264.

[3] L. Adjlout, O. Imine, A. Azzi, M. Belkadi. Laminar natural convection in an inclined cavity with a wavy wall. International Journal of Heat and Mass Transfer 45 (2002) 2141–2152.

[4] Abdalla Al-Amiri, Khalil Khanafer, Joseph Bull, Ioan Pop. Effect of sinusoidal wavy bottom surface on mixed convection heat transfer in a lid-driven cavity. International Journal of Heat and Mass Transfer 50 (2007) 1771–1780.

[5] B.V. Rathish Kumar, Shalini. Free convection in a non-Darcian wavy porous enclosure.

International Journal of Engineering Science 41 (2003) 1827–1848.

[6] M. K. Moallemi and K. S. Jang. Prandtl number effects on laminar mixed convection heat transfer in a lid-driven cavity. HF. J. Hear Mass Transfer. Vol. 35, No. 8 (1992) 1881-1892.

[7] Ercan Erturk and Bahtiyar Dursun. Numerical solutions of 2-D steady incompressible flow in a driven skewed cavity. No. 5 (2007) 377-392.

[8] Hiroyuki Ozoe, Hayatoshi Sayama Stuart and W. Churchill.Natural convection in an inclined rectangular channel at various aspect ratios and angles-experimental measurements. Int. J. Heat Mass Transfer. Vol. 18 (1975) 1425-1431.

[9] Arnone, A., Liou, M. S. and Povinelli, L. A., “Integration of Navier-Stokes equations using dual time stepping and a multigrid method,” AIAA J., Vol. 33, No. 6 (1995)985-990.

[10]Guoliang He and Yinnian He. The finite volume method based on stabilized finite element for the stationary Navier–Stokes problem. Journal of Computational and Applied Mathematics 205 (2007) 651 – 665.

[11]Xavier Core, Philippe Angot, Jean-Claude Latche. A multilevel local mesh refinement projection method for low Mach number flows. Mathematics and Computers in Simulation 61 (2003) 477–488.

[12] J.P. Hartnett and W.J. Minkowycz. Aiding and opposing mechanisms of mixed convection in a shear-and buoyancy-driven cavity. Int. Comm.Heat Mass transfer, Vol.26, Vol. 7 (1999) 1019-1028.

[13] Choi, Y. H. and Merkle, C. L., “The Application of Preconditioning in Viscous Flows,” Journal of Computational Physics, Vol. 105, No. 2, 1993, pp. 207-223.

[14]Weiss, J. M. and Smith, W. A., “Preconditioning Applied to Variable and Constant Density Time-Accurate Flows on Unstructured Meshes,” AIAA Paper 94-2209, June 1994.

[15] Edwards, J. R. and Liou, M. S., “Low-Diffusion Flux-Splitting Methods for Flows at All Speeds,” AIAA Journal, Vol. 36, No. 9, 1998, pp. 1610-1617.

[16] Yang, Y. L., Lin, Y. C., “Numerical Investigation of Natural Convection in an Inclined Wavy Cavity Using a High-order Differential Scheme,” 2010 航太學會學術研討會，論文編號 C2-2, 2010, 開南大學。

[17] Yang, Y. L., Lin, C. C., “A Mixed Order of Galerkin Discretization for Steady Compressible Navier-Stokes Equations in Lid-Driven Cavity Flows,” 中國機械工程學會第二十五屆全國學術研討會論文集,2009,論文編號：A06-17，大葉大學。

附錄 A

U u

v E ρ ρ ρ ρ

 

 

=  

 

 

，

( / )

c

U uU p

E vU

U E p ρ ρ

ρ

ρ ρ

 

 + 

 

=  

 + 

 

，

( / )

c

V F uV

vV p V E p

ρ ρ ρ

ρ ρ

 

 

=  + 

 + 

 

，

0 +

xx

v xy

xx xy

E

u v K T x τ

τ τ τ

 

 

=  

 

 + ∂ 

 ∂ 

 

，

0 +

xy

v yy

xy yy

F

u v K T y τ

τ

τ τ

 

 

=  

 + ∂ 

 ∂ 

 

其中：

ρ

為密度，

u

與

v

別為

x

、

y

方向之速度，

E

為總內能，

p

_為靜壓

,

U V

為

x

、

y

方向之相對速度

t , t

U = − u x V = − v y

2 2

2 3 3

xx

u u v

x x y

τ = µ ^∂ ∂ − µ ^   ^∂ ∂ + ^∂ ∂ ^   − ρκ

2 2

2 3 3

yy

v v u

y y x

τ = µ ^∂ ∂ − µ ^   ^∂ ∂ + ^∂ ∂ ^   − ρκ

2 xy 3

u v y x

τ = µ ^   ^∂ ∂ + ^∂ ∂ ^   − ρκ

x y _t , _t

_{為延著網格之速度，}

µ

為

µ + µ _t

而

µ

為層流之黏滯係數

µ _t

為紊流之黏滯係數，

K

為熱傳導係數，

T

為溫度。

附錄 B-1

ρ U ρu

ρv ρE

 

 

=  

 

 

，

ξη ξη

x x J = y y

( )

c

q qu y p E qv x p

q E p

η η

ρ ρ

 

 + 

 

′ =  − 

 

+

 

 

，

( )

c

u y

F v x

E p

ξ ξ

ργ ργ ργ γ ρ

 

 − 

 

′ =  − 

 

+

 

 

1 2

0 v

x

E

u v Q

τ τ

 

 

′ =  

 + − 

 

，

3 4

0 v

y

F

u v Q

τ τ

 

 

′ =  

 + − 

 

 

ξ ξ

η η

γ uy vx

vx uy

q

+

−

=

−

=

η ξ

η η

x q y

q Q

x q y

q Q

y x

y

y x

x

−

=

−

=

( )

(

_ξ _η _η _ξ

)

η η

ξ

x T x

J T q K

y T y

J T q K

y x

+

−

=

−

=

ξ ξ

η η

τ τ

τ

τ τ

τ

τ τ

τ

τ τ

τ

x y

yy xy

xy xy

yy xy

xy xx

+

−

=

+

−

=

−

=

−

=

4 3 2 1

( )

t xy

yy xx

v y v

y u

x u

J x

u y u

y v

x v

J x

v x v

x u

y u

J y

µ µ µ τ µ τ µ τ µ

η ξ ξ

η η

ξ ξ

η

η ξ ξ

η η

ξ ξ

η

η ξ ξ

η η

ξ ξ

η

+

=

− +

+

−

=

+

− +

−

=

− +

−

=

2 2

3 2 2

2 2

3 2

2 附錄 B-2

1

2

0 0 0

0 sin sin

I

x y

Q u

v p

q y x

q y

A x

q

py px q

y x

y

A x

py px

U a b

U a

Z b

U

pa pb U

U q k k

a y

η η

η

η η

ξ ξ

ξ

ξ ξ

ρ

ρ ρ

γ γ

γ ρ ρ

γ ρ

γ γ γ

ρ ρ

ρ ρ γ γ ρ

γ

   

=  

   

 

 − 

 

=   −  

 

 − 

 

 − 

 − 

 

=  

 

 − 

 

 

 

=  

 

 

= +

= sin sin

sin sin

x y

k y k

b x k x k

η ξ

−

= − +

( )

( ) ( ) ^{( )} ( )

0 0 0 0

4 1

0 2 0

3 3

1 4

0 2 0

3 3

1 1

2 0 0 2

y y x x y y x x

J J

T y x x x x x y y

J J

pK K

y y x x y y x x

RJ RJ

η ξ η ξ η ξ η ξ

µ µ

ρ ρ

µ µ

ρ ρ

γ γ

 

   

 ⋅  −  ⋅ + 

   

 

= ⋅ + ⋅  −  

 

− − −

 ⋅ − − ⋅ − − 

 

 

( ) ( )

( ) ( ) ⁽ ⁾ ( )

2 2

2 2 2 2

0 0 0 0

4 1

0 0

3 3

1 4

0 0

3 3

1 1

0 0

y x y x

J J

S y x x y

J J

pK K

y x y x

RJ RJ

η η η η

µ µ

ρ ρ

µ µ

ρ ρ

γ γ

ρ ρ

 

   

 ⋅ +  − ⋅ 

   

 

= − ⋅ ⋅ +  

 

− − −

 ⋅ + × + 

 

 

( ) ( )

( ) ( ) ⁽ ⁾ ( )

2 2

2 2 2 2

0 0 0 0

4 1

0 0

3 3

1 4

0 0

3 3

1 1

0 0

y x y x

J J

S y x x y

J J

pK K

y x y x

RJ RJ

ξ ξ ξ ξ

µ µ

ρ ρ

µ µ

ρ ρ

γ γ

ρ ρ

 

 

 

 ⋅ +  − ⋅ 

   

 

= ⋅ ⋅ +  

 

− − −

 ⋅ + × + 

 

 

表一、瑞利數 10

³

不同網格分析結果比較。

Ref. [2] result result Difference (%) Difference (%)

Ra=10

³

41×41 81×81 41×41 81×81

Umax 3.649 3.6330 3.6468 -0.44 -0.06 Y@Umax 0.813 0.8127 0.8130 -0.04 0

Vmax 3.697 3.6871 3.6876 -0.27 -0.25 X@Vmax 0.178 0.1887 0.1820 6.01 2.25

Nu

max 1.505 1.5064 1.5070 0.09 0.13 Y@Numax 0.092 0.0918 0.0889 -0.22 -3.37

Nu

min 0.692 0.6937 0.6917 0.25 -0.04

Y@Nu

min 1 1 1 0 0

Nu

avg 1.118 1.1178 1.1179 -0.018 -0.009

表二、瑞利數 10

⁴

不同網格分析結果比較。

Ref. [2] result result Difference (%) Difference (%)

Ra=10

⁴

41×41 81×81 41×41 81×81

Umax 16.178 16.0255 16.1541 -0.96 -0.15 Y@Umax 0.823 0.8229 0.8230 -0.01 0

Vmax 19.617 19.4421 19.5763 -0.89 -0.21 X@Vmax 0.119 0.1231 0.1172 3.45 -1.51

Nu

Nu

avg 4.519 4.5311 4.5163 0.27 0.06

表四、瑞利數 10

⁶

不同網格分析結果比較。

Ref. [2] result result Difference (%) Difference (%)

Ra=10

⁶

41×41 81×81 41×41 81×81

Umax 64.63 64.9900 64.7265 0.56 0.15 Y@Umax 0.850 0.8602 0.8523 1.20 0.27 Vmax 219.36 213.8130 218.6860 -2.53 -0.31 X@Vmax 0.0379 0.0356 0.0390 -6.07 2.90

Nu

max 17.925 17.9934 17.8029 0.38 -0.68 Y@Numax 0.0378 0.0312 0.0363 -17.46 -3.97

Nu

min 0.989 0.9515 0.9821 -3.79 -0.70

Y@Nu

min 1 1 1 0 0

Nu

avg 8.800 8.8416 8.8187 0.47 0.02

表五、瑞利數 10

³

不同溫差之結果比較。

Ra=10

³

∆ =

T

∆ = T 10 ∆ = T 100 ∆ = T 300

Umax 3.6468 3.6456 3.6212 3.4847 Y@Umax 0.8130 0.8120 0.8039 0.7833 Vmax 3.6876 3.6801 3.5786 3.4307 X@Vmax 0.1820 0.1832 0.2073 0.2753

Nu

max 1.5070 1.5031 1.4757 1.4261 Y@Numax 0.0889 0.0873 0.0659 0.0262

Nu

min 0.6917 0.6958 0.7370 0.8040

Y@Nu

min 1 1 1 1

Nu

avg 1.1179 1.1177 1.1174 1.1147

表六、瑞利數 10

⁴

不同溫差之結果比較。

Ra=10

⁴

∆ =

T

∆ = T 10 ∆ = T 100 ∆ = T 300

Umax 16.1541 16.2065 16.3229 16.2041 Y@Umax 0.8230 0.8225 0.8122 0.7892

Vmax 19.5763 19.6121 19.2758 18.6206 X@Vmax 0.1172 0.1163 0.1377 0.1600

Nu

max 3.5326 3.5437 3.5797 3.6983 Y@Numax 0.1439 0.1436 0.1367 0.1160

Nu

min 0.5861 0.5938 0.6432 0.7463

Y@Nu

min 1 1 1 1

Nu

avg 2.2425 2.2493 2.2459 2.2629

表七、瑞利數 10

⁵

不同溫差之結果比較。

Ra=10

⁵

∆ =

T

∆ = T 10 ∆ = T 100 ∆ = T 300

Umax 34.6900 35.0306 37.4377 41.4199 Y@Umax 0.8550 0.8547 0.8513 0.8404

Vmax 68.2023 68.5300 68.9870 68.9211 X@Vmax 0.0653 0.0632 0.0717 0.0895

Nu

max 7.7567 7.7561 8.0009 8.7101 Y@Numax 0.0797 0.0815 0.0812 0.0755

Nu

min 0.7291 0.7335 0.7718 0.8418

Y@Nu

4.404 4.403 4.413

表九、平均紐塞數與文獻結果之比較(Gr=10

⁴

、Re=1000)。

Nu

^N=0 ^N=1 ^N=2 ^N=3

Al-Amiri[7] 7.353 7.353 8.080 7.219 2

^nd

order FV 6.6308 6.7905 7.4604 6.6725

表十、平均紐塞數與文獻結果之比較(Gr=10

⁴

、Re=100)。

Nu

^N=0 ^N=1 ^N=2 ^N=3

Al-Amiri[7] 3.097 3.175 3.253 3.234 2

^nd

order FV 2.7757 2.8486 2.9222 2.8921

圖 2-1 剪切流之正方形孔穴示意圖。

圖 2-2 自然對流之正方形孔穴計算格點與傾斜角度示意圖（81 x 81）。

圖 2-3 混合對流之正方形孔穴示意圖。

u = v = 0 a d i a b a t i c x - m o n .

u = v = 0 a d i a b a t i c x - m o n . u = U

top

, v = 0 , T = T

top

, y - m o m .

u = v = 0 , T = T

bot

, y - m o m . x

y

A φ

T

(a)

(b)

圖 4-1（a）網格 41x41， =0.035

β

。（b）為網格

81x81

， =0.025

β

。

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

mesh 41x41 mesh 81x81 Ghia et al.

Re100

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

mesh 41x41 mesh 81x81 Ghia et al.

Re400

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

mesh 41x41 mesh 81x81 Ghia et al.

Re1000

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

mesh 41x41 mesh 81x81 Ghia et al.

Re3200

圖 4-2 不同雷諾數，水平速度分佈圖，實線(81x81)與虛線(41x41)為實驗結果，圓點為參考文獻結果。

(a)

-0 .0 2 -0 .0 1

-0 .0 0

0.0 1 0.0 2

-0 .0 1 -0 .0 1

(b)

-0 .0 8 -0 .0 6 -0 .0 4

-0 .0 2

-0. 02 0.0 0

-0.02

圖 4-6 Re=3200，不同馬赫數下之密度分佈，(a) M=0.5 (b) M=0.8。

(a)

0.32 0.20

0.12 0.08

0.04 0 .1 6

0 .2 0

(b)

0 .2 8

0. 04 0. 12 0. 20

0.28 0.36

0 .2 8

圖4-7 Re=3200，不同馬赫數下之馬赫數分佈，(a) M=0.5 (b) M=0.8。

(a) ^X

0 0.2 0.4 0.6 0.8 1

-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06