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

286

3

hydrodynamic forces are used to determine hydrodynamic coefficients for the corresponding matrix elements in equation (1). The hydrodynamic force is determined by the pressure distribution p on moving profile surface, while pressure is associated with flow potential Ф . Flow potential determination. Let’s analyze flow near a system consisting of N identical circular profiles in the approach flow (recall that each of profiles is a corresponding tube cross section by plane z = const) fig. 1.

Fig.1. Calculation scheme for hydrodynamic interaction determination.

The approach flow velocity from infinity is V 0 . Each of the profiles is impenetrable and can move with speed i V  Then flow potential Ф must satisfy the following equations and boundary conditions, as fallows from Nikolaev and Smirnov (1985): 0,   (2)

  

V

V и 

i

N

cos

sin

,

1,...,

,



i 



(3)

ix

i

iy

r r a

i

i

. 2 2    x y

V x 0   when

(4)

Calculating  from (2) to (4), even for a special arrangement of a small number of profiles ( N ≥ 2), there are difficulties associated with the complexity of flow region boundary structure. Hydrodynamic forces acting on impermeable profiles can be found by obtaining a potential value only on the contour of each profile using method that allows us to approximate the flow potential nearby each profile. Determine:    i if r a i N i , 1,...,   , (5) for further it is important to assume that i j i j N е R a ij ij , , 1,..., 1,     . (6) Anyway 1 2  ij е , and for real bundles 1 3.  ij е Let:   , max ij е е  (7) where е small parameter. Flow potential in the surroundings of the i- th contour will be calculated as following: ..., 2 1 0         i i i i (8)