A Numerical Analysis of the Behavior of the Free Surface in a Moving Cup

기호설명

  : [m/

]

 : [m/

]

 : volume fraction

  : [m/

]

 : [m/

]

 : []



:  []



: []

 : []

 : []

책임저자 회원 경희대학교 기계공학과, ,

E-mail : [email protected]

TEL : (031)201-2869 FAX : (031)201-2869

*

**

 : [ ]



:

  : [ ]



: (  )



: (  )



: (  )

 :



:

 : [⋅ ]

 : [

 ]



: [⋅

]

1. 서 론

,

홍태협

채희문

김창녕

A Numerical Analysis of the Behavior of the Free Surface in a Moving Cup

Tae Hyub Hong , Hee Moon Chae^* and Chyang Nyung Kim^**

Key Words:

Dimensional Analysis ( 차원해석 ), Free Surface ( 자유 표면 ), Moving-cup ( 움직이는 컵), VOF (Volume of Fluid)

Abstract

A manipulator is operated for the motion of mechanical hands or arms by mechanical mechanism. When

a cup including liquid inside is shifted by a manipulator, it is important to know how a free surface of the

liquid moves. In this study, non-dimensional parameters have been found that affect the rise of the free

surface in a cup moving with constant acceleration. The non-dimensional parameters are the dimensionless

time, the ratio of inertia effect to vicous effect (Reynolds number), aspect ratio of the liquid inside the cup

and acceleration ratio (Froude number). Through this study, the height of the free surface rise in a cup has

been predicted. Generally the maximum rise of the free surface is dependent on the Reynolds number and

Froude number strongly, but on the aspect ratio weakly. But the influence of the aspect ratio on the

maximum rise of the free surface in not negligible in the range 10 < Re < 100.

.

.

,

.

.

.

.

, 3

.

2. 연구방법

무차원 수의 유도 2.1

자유표면을 갖는 액체의 거동 특성을 파악하기 위하여 자유 표면의 거동에 영향을 미치는 인자 들이 파악되었고 그 인자들은 ,  , , ,   , ,

,   , 이다. 이와 같이 자유 표면의 상승에 미치 는 인자가 많으므로 차원해석을 통해 각 변수들 을 무차원화하고 각 무차원 수들이 미치는 영향 을 분석하는 것이 효율적이다.

Buckingham pi theorem

()

(1) .

 

     ^ 

 ^  

  





  

   

  (1)

(1)

. 

(normalized

level rise) .

 ^    ^ _{ , }

_



, 

, 

(aspect ratio) . 





 Reynolds

Froude . Reynolds 



,

Froude 





 ^ _{ ,} 

 Reynolds

Froude . (1)



 maximum normalized level rise .

수치 해석 2.2

수치 해석 모델 2.2.1

. Maximum normalized level rise

.

,

.

, , ,

. ,

,

Fig. 1 .

-  

  

.

Fig. 2 0.035

m, 0.07 m

24,000 .

(center)

.

normalized level rise (Re, aspect ratio, Fr)

.  



= 1 

, 

= 0.035  .  

Re  ,

aspect ratio  .

Fr  .

,  , 

Table 2, Table

3, Table 4 .

2.2.2

, Navier-

Stokes .

, Navier-Stokes .

 ∂

∂  ∇⋅     (2)

 

    ∇⋅    

  ∇  ∇⋅ 

      

(3)

(3) 

, 

(4) .



 ∇   ∇ 



 (4)

1/1000

.

VOF

method .

 .

,  volume fraction

( )

.  0 1

.  .

 ∂

∂  ∇⋅      (5)

 





 × 



 × 



 × 



 × 



 × 



 × 



 × 



 × 



 × 

Table 1 Reynolds numbers with the change of the kinematic viscosity of the engine oil (

= 1 

, 

= 0.035 )

Fig. 1 Schematic illustration of the free surface of a liquid in a cup

Fig. 2 Computational grid system for the liquid and air

in a cup

 











Table 2 Aspect ratios with the change of the initial height of the engine oil (

= 1 

, 

= 0.035 )

 

 Froude number









Table 3 Froude numbers with the change of the acceleration on the engine oil (

= 1 

, 

= 0.035 )

0

,

.

90° .

∆ 0.005

, 



.

CFD

CFD -ACE2006 .

2

(second order upwind scheme) , SIMPLEC

.

연구 결과 및 고찰 3.

컵안에서의 엔진 오일의 유동 특성 3.1

3.1.1 무차원 시간에 따른 엔진 오일의 유동 특성 Fig. 3

.

 ^    ^ 

(normalized level rise (h/R))

. normalized level rise

.

maximum normalized level rise , maximum normalized level rise Reynolds

.

3.1.2

Froude 0.1 Reynolds aspect ratio maximum normalized level rise

3 Fig. 4 . 3

grid

. Reynolds

100 Reynolds

maximum normalized level rise

100 maximum

normalized level rise

. aspect ratio Re > 100

Reynolds maximum normalized

level rise . ,

Reynolds ( Re = 5)

( Re = 1200)

aspect ratio maximum normalized

level rise . , ,

Reynolds 25 aspect ratio

maximum normalized level rise

 

Fig. 3 Change of the height of the free surface at the left

wall ( Fr = 0.20 , Aspect ratio = 0.86 )

( aspect ratio 0.6

) . aspect ratio

0.6 maximum normalized level rise .

.

. aspect ratio maximum

normalized level rise .

Reynolds ,

aspect ratio

.

Reynolds 528 Froude aspect

ratio maximum normalized level

rise Fig. 5 .

Froude Froude

maximum normalized level rise . aspect ratio maximum normalized level rise

. aspect ratio

.

동역학적 상사 여부 3.2

Normalized level rise

. normalized level rise

(dynamic similarity) . Table 5, Table

6 Reynolds

(combination) (Case)

. (1)

,  , ,  ,

 . Table 5 Table 6

Reynolds , Froude , aspect ratio .

    































Table 4 Three cases of A for Re = 528 with Fr

= 0.10 ,



= 0.86

    































Table 5 Three cases of B for Re = 26 with Fr = 0.10 ,



= 0.86

Fig. 4 Effect of the Reynolds number and the aspect ratio on the maximum normalized level rise with Froude numbers = 0.10

Fig. 5 Effect of the aspect ratio and the Froude

number on the maximum normalized level

rise with Reynolds numbers = 528

Fig. 6 Fig. 7

Case A, Case B(Table 5, Table 6

) .

, normalized

level rise .

Case A-1, Case A-2, Case A-3 , Case B-1, Case B-2, Case B-3

. Table 5 6

,

, , ,

Reynolds , Froude , aspect ratio

Fig. 5, Fig. 6

.

4. 결 론

.

.

, normalized level rise , Reynolds , Froude , aspect ratio

.

. Reynolds 100

Reynolds maximum

normalized level rise 100

. aspect ratio maximum normalized level rise aspect ratio 0.6

. Froude maximum normalized level rise Froude

. maximum normalized level rise Reynolds Froude

aspect ratio . Reynolds

10 ~ 100 maximum normalized

level rise aspect ratio .

.

.

참고문헌

[1] Donald F. Young, Bruce R. Munson Theodore H. Okiishi "A Brief Introduction to Fluid Mechanics" , Willy

[2] Hirt. C.W. and Nicholls, B.D., " Volume of

fluid(VOF) method for the dynamics of free

boundaries", Journal of Computational Physics,

1981, Vol. 39, pp. 201-225