A CCURACY AND C ONVERGENCE OF THE L OCAL P RECONDITIONING ON THE H IGH A SPECT R ATIO G RIDS

決ࣜ沲ࣜ汆ࣨ^࣭ࣜ卆ࣜ氪ࣜ柣ࣨ^࣮ࣜ劒ࣜ沫ࣜ笇^ࣦ࣯

A CCURACY AND C ONVERGENCE OF THE L OCAL P RECONDITIONING ON THE H IGH A SPECT R ATIO G RIDS

J.E. Lee,

Y. Kim

and J.H. Kwon

The local preconditioning method has both robust convergence and accurate solutions by using local flow properties for parameters in the preconditioning matrix. Preconditioning methods have been very effective to low speed inviscid flows. In the viscous and turbulent flows, deterioration of convergence should be overcame on the high aspect ratio grids to get better convergence and accuracy. In the present study, the local time stepping and min-CFL/max-VNN definitions are applied to compare the results and we propose the method that switches between two methods. The min-CFL definition is applied for inviscid flow problems and the min-CFL/max-VNN definition is implemented to viscous and turbulent flow problems.

Key Words : (Local Preconditioning) High Aspect Ratio Grid in-CFL in-CFL Definition), mn-CFL/max-VNN in-CFL/max-VNN Definition)

* Corresponding author, E-mail: jhkwon@kaist.ac.kr

1.

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[4].

(stagnation point) (singularity)

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min-CFL /max-VNN š«

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‘switching' min-CFL /max-VNN

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10% bump ,

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(1) ª ^{, ,}

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(flux) . Choi Merkle

[1] Weiss Smith

(preconditioning matrix)[2] ġ .

(6) Ħ (parameter) Ɓ

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(limiter)

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2.2

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[1]. ,

3 MUSCL (Monotone Upstream-

centered Schemes for Conservation Laws) ,

[8].

DADI (Diagonalized Alternate Directional Implicit) [3,9-11] .

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loosely coupled algorithm .

[7] .

3.

3.1

,

[3,7,8].

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[12,13].

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3.3 Min-CFL/max-VNN

. min-CFL

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[1]. Choi Merkle

[12,13] .

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ì ćÞĸ à Îß â Ɓ^ÏîƔ^Ï ĸÞĸ à Îß ƛ

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š«_{á ć} ĢƗ

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ƓĢƖ ì «ƃ_ĢƗá ćŃ

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. (13)

¯§§ Von Neumann Number .

(14)

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. CFL VNN

. (10) (12)

, š«

CFL

VNN . (12)

min-CFL

(12) min-CFL/max-VNN .

3.4 Switching

10% bump min-CFL/max-VNN

Fig. 2 Pressure isolines of the original definition when M=0.1 : AR=5, 50, 500, 5000 (from the top).

(a) AR = 5

Fig. 1 (b) AR = 500

Iteration L2normof∆Q

1000 2000 3000

10^-14 10^-12 10^-10 10^-8 10^-6 10^-4 10^-2

Original∆t, AR = 5 Original∆t, AR = 50 Original∆t, AR = 500 Original∆t, AR = 5000

∞

AR = 5000

AR = 500

AR = 50

AR = 5

Fig. 3 Convergence histories of the original definition at AR from 5 to 5000, M=0.1

[1,13] ,

[12]

min-CFL/max-VNN switching . ,

min-CFL/max-VNN .

(12) switching

10%

.

4.

10% bump

(original), min-CFL (Min-CFL),

switching (present)

. ,

2 NACA 4412

(original), min-CFL/max-VNN (Buelow), switching (present)

.

, bump

.

4.1 10% bump

Fig. 1 . 65×71 .

CFL 5 bump

4 . 0.1

0.001 š« 5, 50, 500, 5000

. š«

š« ^{5000 bump 2200}

. 0.001 0.1

0.1 .

Fig. 2 š«

,

. š«

5 5000

. ,

.

Fig. 3 5 š« ^,

Fig. 5 Convergence histories of the switching definition at AR from 5 to 5000, M=0.1

(a) AR = 10

Fig. 6 (b) AR = 10⁵ Iteration

L2normof∆Q

1000 2000 3000

10^-14 10^-12 10^-10 10^-8 10^-6 10^-4 10^-2 10⁰

Min-CFL∆t, AR = 5 Min-CFL∆t, AR = 50 Min-CFL∆t, AR = 500 Min-CFL∆t, AR = 5000 M_∞= 0.1

AR = 5000

AR = 500 50

5

Fig. 4 Convergence histories of the min-CFL definition at AR from 5 to 5000, M=0.1

min-CFL , switching

. š« ^{5 50 Fig. 3}

Fig. 4 min-CFL

. š« ^{500 5000}

min-CFL . Fig. 3

Fig. 4 š«

min-CFL š«

[1,12,13].

switching

š« min-CFL

Fig. 3 Fig. 4 Fig. 5 .

4.2

Fig. 6 .

97×65 Fig. 7

Ɩ á Ï .

CFL 5 VNN 100 . 4

0.1, 4×10³, 4×10⁴, 4×10⁵, 4×10⁶, 4×10⁷ .

š«á10, 100, 1000, 10⁴, 10⁵ ,

min-CFL/max-VNN , switching .

Fig. 7

š« ^4×104(100), 4×10⁵(100), 4×10⁵(1000) .

.

Fig. 8 š« ^4×103(10),

4×10⁴(10), 4×10⁵(10)

min-CFL/max-VNN .

š«

.

Fig. 9 š« ^4×104(100),

Fig. 7 Accuracy of the original definition near the wall, Re_’ = 4×10⁴(100), 4×10⁵(100), 4×10⁵(1000)

Fig. 8 Convergence comparisons between the original and min- CFL /max-VNN definitions, Re’ = 4×10³(10), 4×10⁴(10), 4×10⁵(10)

Fig. 10 Accuracy of the original definition near the wall, Re_’ = 4×10⁶(10⁵), 4×10⁷(10⁵).

Fig. 9 Convergence comparisons among the original, min-CFL/

max-VNN, present definitions, Re’ = 4×10⁴(100), 4×10⁵(100)

4×10⁵(100) min-CFL/max-VNN switching

. š«

switching

.

š«

.

4.3

k-w WD+

Fig. 10 .

.

.

Fig. 11 š« 4×10⁶(10⁴),

4×10⁷(10⁴) min-CFL/max-VNN .

š« min-CFL/max-VNN

.

Fig. 11 Convergence comparisons between the min-CFL/max -VNN and present definitions, Re’ = 4×10⁶(10⁴), 4×10⁷(10⁴).

Fig. 12 Convergence comparisons between the min-CFL/max- VNN and present definitions, Re’ = 4×10⁶(10⁵), 4×10⁷(10⁵).

switching min-CFL/max-VNN

.

Fig. 12 š« ^4×106(105),

4×10⁷(10⁵) min-CFL/max-VNN

. š«

min-CFL/max-VNN .

5.

.

, min-CFL ,

min-CFL/max-VNN , switching

,

. switching

š« , š«

min-CFL/max-VNN .

š«

. NACA 4412

.

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