Trans RINA, Vol 161, Part A4, Intl J Maritime Eng, Oct-Dec 2019


 4 x


 


  


 z


42+ EI EI x


+ ( +


  


+ +


 ff   q


q t


+ ( 1)− + = t


q


22  z t


q zt zt and qt qt (0, ) 0 , (1, ) 0 ,


== 


 qt x


2


== 


 qt x


2


(1, ) 2


(0, ) 2


0 , 0


Besides, a random noise with an amplitude of order O(10-3) is applied to the fluid variable q for the initial conditions (Violette, et al, 2007):


zx and q x O(10 ) , ( ,0)


== 


−3 qx


( ,0) t


0 (9b)


It has to be mentioned that θ is the angle between the pipeline and the direction of gravity, and the slope of the seabed or the inclination angle of the pipeline is assumed 


to be γ, and =− .


 2


3. INTEGRAL TRANSFORM SOLUTION


In this section, the coupling system, i.e. the initial- and boundary-value problem given by Equations (8) and (9) are solved through GITT. This method is a semi-analytical method, which is a classical approach for solving heat and fluid flow problems, and can realize controlled accuracy and efficient computational performance (Cotta, 1993; Cotta, 1994; Cotta, 1997 and Cotta, 1998). The application


©2019: The Royal Institution of Naval Architects


Yk ,,(0) 0 d x


Yk (1) 0 ,


== ==


d (0) 2


2


d (1) 2


2 Yk d x Yk 0 0


whereXand i are the eigenfunction and the eigenvalue of problem Equation (10a); likewise Y and k


eigenfunction and the eigenvalue of problem Equation (10b). The eigenfunctions both satisfy the following orthogonality, 1


k  Y x Y x x N= kl k 0 k ( ) ( )dl


where ij and kl kl 0


 X x X x x N= ij i 1


0 i ( ) ( )dj (12a) (12b)


 =ij 0 ; and for ij  =ij 1 . Likewise, for kl  = ; for kl


= ,  = . kl 1 A-325


 is the Kronecker delta. For ij = ,


 ,  ,


 are the (11b) ( ,0) 0 ,


== 


zx


( ,0) t


0 (9a) (8c) (0, ) 0 , (1, ) 0 ,


== 


 zt x


2


== 


 zt x


2


(1, ) 2


(0, ) 2


0 , 0 (8b)


of GITT has been further adopted in the area of structural mechanics (Ma, et al, 2006; An & Su, 2011; Matt, 2013a; Matt, 2013b; An & Su, 2014a; and An & Su, 2014b). The implementation of this technique to solve coupled fluid and structure problems has been more frequent over the last few years (Matt, 2009; Gu, et al, 2012; Gu, et al, 2013a; Gu, et al, 2013b; An & Su, 2015; An, et al, 2016; Li, et al, 2016; and Gu, et al, 2016).


The first step in applying GITT is to define the auxiliary eigenvalue problem. For the transverse displacement of a pipeline and the wake variable, the eigenvalue problems are chosen respectively as:


d ( )Xx X x( ), 0 1 4


4


d x = ii 4


i


d ( )Yx Y x( ), 0 1 4


4


d x = kk 4


k


with the boundary conditions being: 2


X ,,(0) 0 d x


i i X (1) 0 ,


== ==


d (0) 2


Xi


d (1) 2


2 Xi d x 0 0 (11a)   x   x (10a) (10b) () sf


rr L zz gA i


  +m mi gL) sin  q (


mm mm


miU L T L −


2 PAi 2 2 a EImp + )


pp p


 z 


2


(m m gL) cosz EI


2 pi +


tt m


+ + mp


e )


22 22


2 2 + ( x 4 p EID − 3 +


eesin L EID


4 =


2miU  z  x t


2 (8a)


together with the dimensionless boundary conditions expressed as:


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