Algorithms of the Discrete Singularity Method for Computing Technologies
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ALGORITHMS OF THE DISCRETE SINGULARITY METHOD FOR COMPUTING TECHNOLOGIES S. O. Dovgiy,1 S. I. Lyashko,2† and D. I. Cherniy2‡
UDC 517.9; 519.642.7; 532.5
Abstract. Algorithms of the discrete singularity method for computing technologies are considered. The algorithms transform discrete integral representations with discontinuous functions and change the order of singularities in the system of discrete singularities. The results of transformation allow us to correctly calculate the values of the functions and their derivatives under parametric dependence of characteristic functions on time. The algorithms of computing technologies are applicable for both two-dimensional and three-dimensional hydrodynamic problems of non-stationary detached flow. Keywords: discrete vortex method, discrete singularity method, computing technologies. GENERAL PROBLEM SOLUTION In many cases [1–10], to solve a plane problem about non-stationary flow around impermeable, moving r r with velocities Wd and Wu boundary contours Ld ( t ) and Lu ( t ) in a deformable domain D ( t ) , mathematical model with parametrical dependence on time t is used, which in terms of the theory of functions of a complex variable (TFCV) has the integral representations:
F( z, t ) = j ( x, y, t ) + i y ( x, y, t ) =
1 2pi
ò
g ( w, t ) ln( z - w ) dw +
Ld ( t )
1 2pi
ò
g ( w, t ) ln( z - w ) dw ,
(1)
Lu ( t )
¶F( z, t ) ¶z g ( w, t ) dw , z-w
V ( z, t ) = u( x, y, t ) – i u ( x, y, t ) = =
1 2pi
G0 =
ò
Ld ( t )
ò
Ld ( t )
g ( w, t ) 1 dw + 2pi z-w
g ( w, t ) dw +
ò
ò
Lu ( t )
g ( w, t ) dw = const .
(2)
(3)
Lu ( t )
Because of variability of the domain with a priori unknown shape of a part of boundaries, numerical methods can only be used to solve the problem about non-stationary flow around impermeable moving boundaries. 1
Institute of Telecommunications and Global Information Space, National Academy of Sciences of Ukraine, Kyiv, Ukraine, [email protected]. 2Taras Shevchenko National University of Kyiv, Kyiv, Ukraine, †[email protected]; ‡ [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 6, November–December, 2017, pp. 147–159. Original article submitted May 22, 2017. 950
1060-0396/17/5306-0950 ©2017 Springer Science+Business Media New York
DISCRETIZATION OF THE INTEGRAL REPRESENTATIONS So-called vortex method is often used for numerical solution of aerohydromechanic problems [1–10]. It is the discrete singularity method [5], which is based on discretization of the integral representations (1) and (2) according to the conditions of theorems from [2, 4–6]. In this case, we will write the integral representations (1)–(3) (in terms of complex variables) in the domain with piecewise-smooth boundary that admits decomposition into the set of limiting elements L =
M
å Lj
as follows:
j =1
F( z, t ) = j ( x, y, t ) + i y ( x, y, t ) = G j (t )
M
å
=
j =1
2pi
P
n (t )
p
s=1
ln ( z - w 0 j ( t )) + å
å
d sp p ln( z - w 0s ( t )), 2pi
V ( z, t ) = u( x, y, t ) - i u ( x, y, t ) = =
M
G (
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