PSI - Issue 20

Tatiana Fesenko et al. / Procedia Structural Integrity 20 (2019) 284–293 Tatiana Fesenko et al. / Structural Integrity Procedia 00 (2019) 000–000

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7

Thus, it becomes possible to determine flow in the circle of each profile potential with a given accuracy on  parameter. For qualitative analysis it is enough to find forces in the first approximation. 2. Hydrodynamic forces determination Pressure at a point moving with  V velocity relative fixed coordinate system can be expressed in terms of potential as follows Nikolaev and Smirnov (1985):   . 2         V p  (39)

2 1

  t

  

  

1

Taking into account the velocity smallness of oscillating impermeable profiles from (39), we obtain an expression for pressure on i -th profile contour:

    

    

   

   

_

_

_ 2

   

   

~

~

1 a

1 a

2 1

1 a

 

  t

  t

 

 

i

i

i

i

i

, r a i 

p

V

1 



 







when

(40)

i



i 

i 

i 

V V

V

i  sin    ix

cos

i 

azimuth contour points of the i -th profile velocity.

i

iy



Assign three groups of forces acting on each profile. 1. Forces proportional to oscillating profiles accelerations:

~

F F

sin cos

i 

  

  



2

  

  

t  

i

ix

0 

a

d

,



1 

i 

(41)

i 

iy

F ix , F iy – force components on the OX and OY axes. In the linear approximation, we neglect i ~

Ф change due to profiles displacement, then the time derivative of i ~ Ф

in (41) is a value proportional to accelerations of the oscillating profiles. This group of forces determines acceleration connection matrix or added masses matrix. If liquid is stationary ( V 0 = 0), only forces proportional to accelerations will act on profiles. 2. Forces proportional to profiles movement velocities:

   

       

_

_

   

~

S S

sin cos

i 

  

  



2

  

  

 

 

1 a

1 a

t  

i

i

i

ix

0 

a

V

d



.





1 

i 

(42)

i



i 

i 

i 

iy

These forces determine velocity connection matrix. 3. Positional forces depending on profiles relative position (configuration):

   

   

_ 2

P P

sin cos

i 

  

  

2



  

  

a

1 

 

1 a

i

ix

0 

d



.

i 

(43)

i 

2

i 

iy

Positional forces make it possible to determine the stationary (equilibrium) profiles configuration and change in positional forces at deviations from the stationary position determines the matrix of positional relations between the oscillating profiles. We reduce calculations with linear approximation of static deformations.