Numerical Investigation of the Stress-Strain State of the Curved Pipeline

Formulation of the Problem

We consider the pipeline of length L with a circular cross-section of radius 0 R and a wall of small thickness h . The bent pipe axis is the flat curve 0 0 0 0 0 0 0 = { , : = ( ), = ( )} x y x x s y y s  where s is an arc length. The pipe is immersed in a highly viscous medium and filled with a steady flow of a fluid, which is moving with velocity 0 s  under the influence of a constant pressure drop. The geometry of the problem, the coordinate system and the general formulation of the problem are given in [3]. Below is the information necessary to understand the following text. OR  are the polar coordinates in the cross-section of a pipe. All equations of continuum mechanics are written below in curvilinear coordinates according to the methods of [10]. Formulas for all necessary geometrical parameters of the pipe axis and surface are derived in [4,5].

For applicability of model equations, Vlasov's conditions [11] must be valid:

( )

* 0 0 0 = / 0.1; min , / 4. h h R L R  

The following parameter should be small: 𝜆 = 𝑅 0 𝑚𝑎𝑥|𝜅 0 (𝑠)| ≪ 1. For the middle surface of the pipe's wall, the following geometric relationships are executed: ______________________________________________________________________________________ 128 ( ) k are the main curvatures of a median surface,  is the axis curvature. The equations of the three-dimensional mathematical model have next form (see [3,9]). For fluid movement:

0 0 1 2 0 0 ( , ) sin 1 ( , ) sin , ; , 1 ; 1 ( , ) sin st A R s t B R k k R s t R    = + = = = +( ) ( ) ( ) 0 0 0 0 , = , , = 0, = const. f s f v v p v     − −  (1)

( )

1 2 00 00 0 2 = , = , = , = / ; 4 f s s s f f f f R Re R           1 1 0.   =      = +   

For pipe wall movement in a shell approximation:

(0) 2 0 2 12 1 1 1 (1 ) = , k Iw k k u X A s B A s Eh       −  −  − + − − −      (0) 2 0 1 12 1 1 1 (1 ) = , k Iw k k v Y B A s B Eh       −  −  + + − − −      2 (0) 2 2 1 2 1 2 2 1 1 ( ) 2 ( ) ( )12h k k I ABk k w Bk u Ak v w AB s   −    − +  + + + −   −    ( ) 22 2 2 2 12 1 =; 12 h k k w Z Eh  −  −  + −  (2) 2 1 =; BA AB s A s B               +                22 (0) 2 00 1 1 1 = cos sin , 2 u A v A w v I v w A s R A R A s s                   + + + + − +                        ( ) ( ) 0 1 =; 2 Bv Au AB s     −    ( ) 2 0 00 =. 2 t s s R    ( ) 2 0 2 11 =, t t s u X h t h   − +   2* 2 * 00 1 2 cos =, 0.5 ln 4 t e vu Y ht u hR R       −−   −   2 2 11 = ( ), te w Z p p h t h   − + −  * 0 0 * 0 00 2 sin = 1 cos . 0.5 ln 4 ee e R u p gh h u RR       −+    −  

Here denoted: u , v , w are displacements of the median surface of a pipe along the coordinates s ,  , R , respectively; (0) I , 0  are first invariant of the strain tensor and the linear torsion of the pipe wall, respectively; X , Y , Z are components of the density of forces acting on the shell along the coordinates s ,  , R , respectively. For fluid: e  is density of the medium;  is viscosity of medium; 0 h is depth of immersion of the pipeline; * u is velocity of the pipeline's cross-section motion. Equations (2) and the formulas for 0  , 2  are known from the shell theory by V.Z. Vlasov [11].

f  is fluid density inside the pipe; f  is fluid viscosity; s v , v  ,

The system of equations ( 2) is supplemented by the boundary conditions of rigid fixing and by the homogeneous initial conditions:

0, 0, 0 0 w u v w s  = = =  =  at 0 s s L =  = and any t ; 0, 0, 0; 0, 0, 0 u v w u v w t t t    = = = = = =    at 0 t = . (3)

Boundary conditions for the equations of fluid flow (1):

( ) ( ) ( ) 00 0, , = , , , = 0, , , = . s s r a v R v R p L R p     (4)s s s s r r r r v r v r v r v r O v r v r v r v r O v r v r v r v r O p r p r p r p r O                                              = + + +  = + + +  = + + + = + + +(5)

The displacements of the median surface of a pipe wall:

( ) ( ) ( ) 2 01 2 1 2 01 '( , , ) ( ) ( , ) sin ; '( , , )( , ) cos ;

'( , , ) ( ) ( , ) sin .

u u u O v v O w w w O                           = + + =+ = + +(6)

By means of ( 5), ( 6), the task is reduced to a one-dimensional problem. Substitute the expressions ( 5), ( 6) into the three-dimensional problem (1)-( 4). We obtain a reduced problem of the dynamics of the pipeline (see [9]):

22 2 3 2 0 0 0 0 0 2 2 * * 1 =, 2 s u w w w Eh               + − −       (7) 2 24 *2 2 2 4 2 0 0 0 0 0 0 0 2 4 * * 1 = ( ) . 12 2 a s e w w u w h w p L gh Eh                    + + + − + − −          (8) 2 2 2 1 1 1 2 00 1 0 22 1 1 1 2 (1 ) 2 2 2 uw u v w fu u                  − +   − − − + + − + − −       2 2 2 2 22 3 1 1 3 1 0 0 0 0 0 2 2 2 * 2 3 t w w w w R w w u f E                  − + +           (9) * 2 2 1 1 1 1 2 * * * 1 00 4 2 1 1 1 22 0.5 ln | | u e u vu v Eh RR        −  +  − − + +   −   22 2 2 1 1 0 0 0 1 0 *2 3 =; 2 t u w R wv fw w E          −   + + − −       (10) 2 *2 4 2 2 4 1 2 1 1 2 1 0 0 0 11 0 42 2 (1 ) = 12 2 u w w h w w u w f w f wv                            + − + − + + − − +                22 * 2 21 0 1 0 * * * 2 * 1 00 4 2 1 =. 0.5 ln | | t fs u e R u w f E h E RR           −−     −    (11)

As shown in [9], the solution of the problem on fluid dynamics and the solution of the zero-approximation problem ( 7), ( 8) can be considered known. The boundary conditions for the system of the first approximation equations ( 9)-( 11) have the form:

         = =  = =  =   = = =  = = = =   (12)

The difference scheme for solving the initial-boundary value problem ( 9)-( 12) is described in the article [9]. From expressions (6), it follows that the physical sense of 10 uR  is a displacement of points from the cross-sectional plane perpendicular to the axial line of the pipe, i.e., warping [1]. Sectional warping of the cross-section of a cylindrical pipe was observed in the experiments of V.Z. Vlasov in [12]. Displacements of the axial line shown in Figure 1 (a), illustrate the coherence of the numerical calculations for the offered mathematical model with the fundamental laws of mechanics. The existence of the cross-sectional warping is confirmed by the graph of the longitudinal wall displacements in Figure 1 (b), the geometric meaning of which is the warping of the cross-section. Thus, the numerical experiment indicated the existence of the cross-sectional warping of the long thin-walled pipeline of the order 0 0.02 R . In Figure 2, the coordinates of a profile of the pipeline and warping of cross-sections in task 2 are represented. Similar to task 1, the change in the coordinates of the profile at the start (dotted line) and at the end of the calculation in Figure 2 (a) indicates the coherence of the numerical experiment with the laws of mechanics. Cross-sectional warping also occurred in this case, with a warping on the order of 0 0.003R , as shown in Figure 2 In tasks above, the pressure difference is considered to be constant on the ends of a pipe, which enabled the constant velocity of an internal stream of fluid. For all tasks, numerical experiments were conducted in which the pressure difference changed under the piecewise linear law: hours. The results of the displacement calculation and the centreline's coordinates are shown in Figure 3.

  −    + − −  +         +    −    −    + − −        

Figure 3 shows a part of a movement's 3D-graph over a duration of 125 hours. In the graph, the time interval in which the influence of the pressure jump leading to an increase in the deflection is expressed is detached. Next, according to the decrease in the pressure, the pipe returns to the position corresponding to the position of the final equilibrium of task 2.

In Figure 3 (b), the coordinates of the axial line at the beginning of the calculation, at the moment of time corresponding to the strongest deviation of a profile and at the final moment of time are shown. It was established that the slowly varying pressure difference eventually leads to the same steady state as the constant pressure difference, provided that the final values of the differences are equal.