=Paper=
{{Paper
|id=Vol-2426/paper19
|storemode=property
|title=Numerical Investigation of the Stress-Strain State of the Curved Pipeline
|pdfUrl=https://ceur-ws.org/Vol-2426/paper19.pdf
|volume=Vol-2426
|authors=Viktor A. Rukavishnikov,Oleg P. Tkachenko,Anna S. Ryabokon’
}}
==Numerical Investigation of the Stress-Strain State of the Curved Pipeline==
https://ceur-ws.org/Vol-2426/paper19.pdf
Numerical Investigation of the Stress-Strain State of the Curved
Pipeline
Viktor A. Rukavishnikov¹, Oleg P. Tkachenko¹, Anna S. Ryabokon’¹
¹Computing Center of Far-Eastern Branch, Russian Academy of Sciences, Khabarovsk, Russia,
vark0102@mail.ru
olegt1964@gmail.com
anyuta.riabokon@yandex.ru
Abstract
We investigate three-dimensional mathematical model of the pipeline,
which is based on the shells theory by V.Z. Vlasov. A method for the
approximate solution of the model equations based on the asymptotic
expansion of the solution is proposed. Results of numerical experiments
for model problems are given. It is proved that the mathematical model
adequately describes the known phenomena of pipelines mechanics. In
the numerical experiment, the deplanations of the pipe cross sections were
found.
1 Introduction
Presently, the world has begun to work on the development of oil and gas deposits in remote areas. With this
development, many new scientific and technical problems have arisen. One of the unsolved problems is the problem
of rejection of the pipeline from its design position, see in [1, 2]. Next, the problem arises regarding the study of the
dynamics of a pipeline in a deformable medium. A brief review of scientific publications on this topic was made in
[3, 4]. In a series of papers [5–9], the kinematics and dynamics of a bent pipeline were investigated, mathematical
models were constructed, algorithms for their numerical analysis were developed, and test problems were solved.
Main purposes of this paper are to (1) present a new problem transformation algorithm from three-dimensional to
one-dimensional formulation; (2) perform computational experiments on the proposed mathematical model; (3) prove
that the mathematical model describes the deplanation of pipe cross sections and (4) determine the displacements of
the pipeline for a variable internal pressure difference.
2 Formulation of the Problem
We consider the pipeline of length L with a circular cross-section of radius R 0 and a wall of small thickness h .
The bent pipe axis is the flat curve 0 = { x0 , y0 : x0 = x0 ( s ), y0 = y0 ( 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 s 0
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.
There are two coordinate systems: the global Cartesian coordinate system ( O , x, y , z ) and the curvilinear
coordinate system ( O , s , , R ) . Curvilinear coordinates ( O , s , , R ) build on the currently axis so that s is the
length of the arc ( OO ) , and ( O , , R ) 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:
h = h / R0 0.1; min ( L, 0 ) / R0 4.
*
The following parameter should be small:
𝜆 = 𝑅0 𝑚𝑎𝑥|𝜅0 (𝑠)| ≪ 1.
For the middle surface of the pipe’s wall, the following geometric relationships are executed:
Copyright © 2019 for the individual papers by the papers' authors. Copyright © 2019 for the volume as a collection by its
editors. This volume and its papers are published under the Creative Commons License Attribution 4.0 International (CC BY 4.0).
In: Sergey I. Smagin, Alexander A. Zatsarinnyy (eds.): V International Conference Information Technologies and High-
Performance Computing (ITHPC-2019), Khabarovsk, Russia, 16-19 Sep, 2019, published at http://ceur-ws.org
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A = 1 + R0 ( s , t ) sin , B = R 0 ; k1 =
( s , t ) sin , k 2 = 1 R0 ;
(1 + ( s , t ) R0 sin )
where k1 and k 2 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:
f ( v 0 , ) v 0 = − p − ( s 0 ) , ( , v 0 ) = 0, f = const. (1)
1 f 2 s 0 R0
( s 0 ) = s20 , = , Re = , f = f / f ;
4 R0 f
64
1 = , Re 2000;
Re
= 0.0032 + 0.221 , Re 2000.
1
Re 0.237
For pipe wall movement in a shell approximation:
1 I 1 − 0 k w 1 −
(0) 2
− + (1 − ) k1k 2 u − 2 =− X,
A s B A s Eh
1 I 1 − 0 k w 1 −
(0) 2
+ + (1 − ) k1k 2 v − 1 =− Y,
B A s B Eh
1 − h
2
− ( k1 + k 2 ) I (0) + 2 ABk1k 2 w + ( Bk 2 u ) + ( Ak1v ) − 2 2 w −
AB s 12
1 −
2 2
2 ( k12 + k 22 ) w = −
h
− Z; (2)
12 Eh
1 B A
= +
2
;
AB s A s B
1 u 1 A 1 w v
2 2
A v
I
(0)
= + + v cos + + sin w − 2
+
,
A s R0 A R0 2 A s s
1 R0
0 = ( Bv ) − ( Au ) ; t ( s 0 ) = s20 .
2 AB s 2
u 2v 2 u * cos
2
1 1 1
X = −t + t ( s 0 ) , Y = −t − ,
t t
2
eu*
2
h h h
hR 0 0.5 − ln R0
4
w 2 u * sin
2
1 1 R
Z = −t + ( p − p e ), p e = e gh0 1 − 0 cos + .
t
2
h h h0 eu*
R0 0.5 − ln R0
4
Here denoted: u , v , w are displacements of the median surface of a pipe along the coordinates s , , R ,
respectively; I (0) , 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: f is fluid density inside the pipe; f is fluid viscosity; v s , v , v R are components of the fluid velocity vector
v 0 along the coordinate axes s , , R , respectively. For expernal pressure: = 1.7811 - number of Euler-
Maskeroni; e is density of the medium; is viscosity of medium; h0 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].
The system of equations (2) is supplemented by the boundary conditions of rigid fixing and by the homogeneous
initial conditions:
w
u = 0, v = 0, w = 0 = 0 at s = 0 s = L and any t ;
s
u v w
u = 0, v = 0, w = 0; = 0, = 0, = 0 at t = 0 . (3)
t t t
Boundary conditions for the equations of fluid flow (1):
v s ( 0, , R ) = s 0 , v r ( , , R0 ) = 0, p ( L, , R ) = p a . (4)
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3 Applying Transformation Algorithm to Mathematical Model
We introduce dimensionless variables: = s / , r = R / R 0 , = , = t . Similarly, we do for unknown
functions: u = u / R 0 , v = v / R0 , w = w / R 0 ; v s = v s 0 / , v = v 0 / , v r = v R 0 / , p = p / pa , L = L / .
The appropriate dimensionless unknown functions are presented as follows:
Velocity and pressure in fluid:
v s ( , , r ) = v s( 0 ) ( , r ) + v s(1) ( , r ) sin + v s( 2 ) ( , r ) cos + O ( 2 ) ,
v ( , , r ) = v
(0)
( , r ) + v(1) ( , r ) sin + v( 2 ) ( , r ) cos + O ( 2 ) ,
(5)
v r ( , , r ) = v r
(0)
( , r ) + v r(1) ( , r ) sin + v r( 2 ) ( , r ) cos + O ( 2 ) ,
p ' ( , , r ) = p ( 0 ) ( , r ) + p (1) ( , r ) sin + p ( 2 ) ( , r ) cos + O ( 2 ) .
The displacements of the median surface of a pipe wall:
u '( , , ) = u 0 ( ) + u1 ( , ) sin + O ( 2 ) ;
v '( , , ) = v1 ( , ) cos + O ( 2 ) ; (6)
w '( , , ) = w 0 ( ) + w1 ( , ) sin + O ( ) .
2
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]):
u0 w0 3 w0 w0 1
2 2
2 + − 2
=− * * s20 , (7)
2
E h 2
2
h *2 2 w 0
2
4 w0 u 2 w0
4
1
w0 + + + 0 − = * * p a + s 0 ( L − ) − e gh0 .
2
4 (8)
12 2
2 E h
u1 1 − 1 + v1 w1 1 − 2 u0 w
2 2
2 − u1 − + + f u 0 − 2 + (1 − ) 0 −
2 2
2 2 2
3 w 1 w0 w0 2 w 1 w0 w0 t R0 2 u 1
2 2 2 2
− + +
3
2
3 f = ; (9)
2 2 * 2
E
1 − v1 2 u1 1 + u1
2 *
1
2 − v1 − + +
R0 0.5 − ln | R0 |
2 * *
2 E h e u1* 2
4
3 − u0 2 w0 w 1 R 2 v1 2 2
+ w 1 + f w0 − − = t 0* ; (10)
2
2 E
2
h 4 4 w1
*2
2 w1
2
u1 u0 2 w0 w 1 2 w0
w1 + − 2
+ − v 1 + f 2 w + (1 − ) − + f =
2
4 0
12
1 2 u *
t R02 2 2 w 1
= * * f s 0 f − −
2 1
. (11)
2
R0 0.5 − ln | 4e 1 R0 |
E h u * E*
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:
w1
v 1 = w 1 = 0 = 0 at = 0 = L ' and any ;
(12)
u1 v1 w1
u1 = v 1 = w 1 = 0 = = = 0 at = 0 and any .
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 u1 R 0 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].
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4 Numerical Experiments for Mechanical Problems
Task 1. Cubic parabola y = 10 −8 x ( x − 6000 ) ( x − 12000 ) , 0 x 12000 . Values of the main geometrical
parameters: L = 12500 m , min 0 3951.5 m . Uniform current velocity s 0 = 1 m / s calculated over the interval of
time Tend = 864000 s or 10 days.
Task 2. Fractional-rational function y = 40 (1 − 0.001 x ) / (1 + 10 −6 x 2 ) , − 6000 x 6000 . Values of the main
geometrical parameters: L = 12000 m , min 0 1194.5 m . Uniform current velocity s 0 = 1 m / s calculated interval
of time Tend = 691200 s or 8 days.
Other parameters are equal: e = 1700 k g / m 3 , = 10000 N s / m 2 , t = 7200 kg / m 3 , h = 0.005 m ,
E = 2.07 10 N / m , = 0.24 , R 0 = 0.3 m , f = 0.667 N s / m 2 , f = 850 kg / m 3 .
11 2
Figure 1 shows the coordinates of the pipeline profile and its warping in task 1.
Figure 1: (a) – the profile coordinates at the start (dashed line) and at the calculation end; (b) – the longitudinal
displacement in a first approximation.
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 0.02 R 0 .
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 0.003R 0 , as shown in Figure 2 (b).
Figure 2: (a) – the profile coordinates at the start (dashed line) and at the calculation end; (b) – the longitudinal
displacement in a first approximation.
5 The Pipeline Displacements under the Dynamic Pressure Action
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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:
p 0 , t t0 ;
p k + k p − 1 ( t − t − t ) , t < t < t + t;
0 p t
0 0
0
p = k p p 0 , t 0 + t t t1 − t ;
k −1
p 1 + p
( t1 − t ) , t1 − t < t < t1 ;
0
t
0 p , t t1 .
Here, p0 is the pressure difference providing the flow velocity s 0 ; t 0 , t1 – time of the beginning and of the
ending of a pressure's jump, respectively; t – time during which the pressure changes; and k p is factor of the
pressure's increase. The initial pressure drop is determined by the formula for the hydraulic resistance of the pipe
(see [13]):
L
p 0 = 1 f s20 ,
4 R0
with the other parameters set by the formulation of the numerical experiment.
In the example shown here, Tend = 1209600 s or 14 days; k p = 1.5 ; t0 = 86400 s , t1 = 360000 s or 4 days; t = 2
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.
6 Conclusion
We presented the new mathematical model of dynamics of the curvilinear pipeline on the basis of the shells theory.
A method for the approximate solution of the model equations based on the asymptotic expansion of the solution was
proposed. Results of numerical experiments for model problems were given. It was proved that the mathematical
model adequately describes the known phenomena of pipelines mechanics. In the numerical experiment, the
deplanations of the pipe cross sections were found. It is shown that the mathematical model is applicable in the case
of a slow change in the internal pressure in a fluid.
Figure 3: (a) - normal displacements of the axial line; (b) - coordinates of the axial line for t = 10 minutes (the
continuous line), t = 4 days (dotted line), and t = 14 days (dot-dash line).
Acknowledgements
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This research was supported in through computational resources provided by the Shared Facility Center “Data
Center of FEB RAS” (Khabarovsk).
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