Trans RINA, Vol 161, Part A4, Intl J Maritime Eng, Oct-Dec 2019
a slope. For instance, it is reported that the maximum slope is about 22° for the Southern Route, and 25° for the Northern Route of the pipeline Transmed stretching from Algeria to Italy (Drago, et al, 2015). Paidoussis (1998) pointed out that for vertical pipelines vibrating freely, the effect of gravity is non-negligible, suggesting that the influence of gravity is important on the dynamic response of inclined pipelines and thus should be considered. Gan et al. (2015) and Jing et al. (2015) built a mathematical model to study the vibration behavior of an inclined viscoelastic pipe. Results show that the vibration of inclined fluid-conveying pipes demonstrates bifurcation processes. However, in their model, only the internal fluid was considered.
In this work, considering the effect of gravity, a fluid- structural model is proposed to analyze the dynamic behavior of inclined submarine pipelines subjected to internal flow and external current.A thorough understanding of their dynamic behaviors will help the design of pipelines. In Section 2, the governing equation for the vibration of inclined submarine fluid-transporting pipelines is established, and a non-linear wake oscillator is employed to model the vortex shedding behind the pipeline. Section 3 provides the analytical or semi- analytical solutions for the transverse displacement using the generalized integral transform technique (GITT). Case studies, presented in Section 4, include the structural natural frequency,mode shapes, and effect analysis of the inclination and internal fluid on the dynamic response through time history, and frequency analyses. The final results and discussions are concluded in Section 5 of the paper.
2. MATHEMATICAL MODEL
Consider a Cartesian coordinate system of x’- and z’-axes, with its origin at the left end of the pipeline, where x’-axis is in the direction of the gravity. Being rotated counter- clockwise by θ, a new coordinate system of x-, y- and z- axes is sketched, where the y-axis is parallel to the current and orthogonal to x- and z-axes, and z-is the direction along which the pipeline deflects transversely. Taking the free-spanning submarine pipeline as an example, the diagram is illustrated in Figure 1.
z y Ta U x' V
Figure 1: Schematic diagram of a fluid-conveying free- spanning submarine pipeline over a slope.
z' L
Ta x
U
By neglecting the terms of second or higher order, according to the Euler beam approximation for small deformation, the force equilibrium equations are as follows:
− − +m sin
xtf x m A S g
P i i i and
+ − i
(in x-direction)
z U
= 0 (1a)
In the present study, the pipeline is assumed to be elastic, non-deformed, and simple-supported at both ends. The internal fluid inside the pipe travels at a constant velocity U, and the external current flows at a constant velocity V. The pipeline is cylindrical with a constant outer diameter D and inner diameter Di. Its outer cross section area is symbolized as Ae, the inner cross section area Ai, and the inner perimeter is Si. The axial tension and internal pressure are Ta and P, respectively. This model is constrained to cross-flow vibration.
2.1 STRUCTURE MODEL
The forces and moments acting on the fluid and pipe elements δx are analyzed respectively as Figure 2 shows. The internal fluid is assumed to be steady and incompressible. Since the diameter of the pipe is small compared with the wavelength of the disturbances to the fluid particle, its accelerations in x- 2
and z-directions are respectively zero and
+
tx
Uz.
τ stands for the shear force acting on the inner surface of the pipeline, hence τSi represents the friction between the internal fluid and the inner surface. f is the transverse force between the pipe and the internal fluid. rs is the structural damping. g is the acceleration due to the gravity. For unit length of the
pipeline, mi is the internal fluid mass, mp is the mass of the pipeline, and
e M e = C /4
mD is the added mass due to 2
external fluid, where CM is the added mass coefficient. The density of the pipe, the internal fluid and the external current are expressed respectively as ρp, ρi and ρe. Q is the transverse shear force on the pipe element, andMis the bending moment. Fw is the force due to the current in the cross-flow direction,
expressed as FF , where
w L= +
F CD = V
2 Le L 2 e e sin
gA zzr m tt
2 − − f e f = f e 2 is the lift force, and r is the D 2
external fluid added damping, with being related to the drag mean sectional drag coefficient of the structure CD through
=C St D / (4 ) .
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©2019: The Royal Institution of Naval Architects
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