Dynamics of a Flexible Offshore Gas Transmission Riser ...

International Journal of Mechanics and Applications 2014, 4(3): 59-79 DOI: 10.5923/j.mechanics.20140403.01

Dynamics of a Flexible Offshore Gas Transmission Riser with Random Vortices

Patrick S. Olayiwola1,*, Charles A. Osheku2

1Department of Mechanical &Biomedical Engineering, College of Engineering, Bells University of Technology, Ota, Nigeria 2Centre for Space Transport and Propulsion, National Space Research and Development Agency Federal Ministry of Science and

Technology, Abuja, Nigeria

Abstract In this paper, the mud-fluid-structure interaction mechanics for a deep offshore riser subject to internal and circulating external flows is investigated as a boundary value mathematical physics problem. The driving physics is premised on the kinematics of vortex-induced fully-developed compressible flow. For this problem, the deep offshore riser is idealized as a gas-carrying elastic column using Euler-Bernoulli beam theory. Notably, the subjective vortex flow arising from ocean water effect is considered as a shake-off phenomenon propelled by internal and external flows. It is also shown that the stochastic condition is central to the formation of vortex-induced vibration. By applying integral transforms on the resulting boundary value problem, modal vibration responses of the entire system are analyzed for design applications.

Keywords Mud-fluid-structure interaction mechanics, Offshore riser, Circulating external flow, Kinematics of vortex-induced compressible flow, Shake-off phenomenon

1. Introduction Vertical flows in risers are common fluid-transmitting

means in practice, as found used in evacuating products from the ocean deeps and soil strata. Flexible risers are among the common types of production risers. They may be deployed in a variety of configurations, depending on the water depth and environment. Indeed, flexible pipes have traditionally been limited by diameter and water depth, however, they are often less expensive to install and are more tolerant to dynamic loads. Also, where flow assurance is an issue, the flexible risers can be designed with better insulation properties than a single steel riser. With the Euler beam theory, the riser under investigation is idealized as a gas-carrying column having its lower end connected to the buried flowline and the other end via the ocean deeps to the surface, in which the environment is assumed as one that does not agitate flow turbulence. A stochastic condition of vortex formation causes a vortex-induced vibration to occur within the internal flow of the structure, while the absence of flow rotation ensures completion through pressure gradient and, around the beam, the flow is assumed to be inviscid, irrotational and incompressible. The transmitting fluid is assumed to be an ideal gas and single phase. Both the external and internal flows are described by the continuity equation, then the Euler equation for the former, and the energy equation for the

* Corresponding author: [email protected] (Patrick S. Olayiwola) Published online at http://journal.sapub.org/mechanics Copyright © 2014 Scientific & Academic Publishing. All Rights Reserved

latter. On invoking into the structural mechanics equations, the ocean and gas domains equations, the resulted governing differential equations are investigated as we further invoke the integral transforms, for obtaining structural dynamic responses and vibrations.

Really in vertical flows, gas reduces pressure drops at reasonably low mass flow rate /length of riser ratio. Gases have low densities and because of this they contribute to the lightening of pressure gradients in flows through the risers. The problems associated with gas flow in vertical pipes involve the determination of gas density at a particular pressure and temperature, determining the actual volume that a certain quantity of gas will occupy under a set of condition (gas compressibility), and the velocity of a gas in the pipe at a particular condition.

In oil and gas production, vertical gas flow is found in practically every tubing string used. Then, in order to correctly select completion string, predict flow rates, and design required lift installations, it is necessary to be able to predict a vertical flow pressure traverse. The pressure gradient (that is, the rate of change of pressure with respect to unit of flow length) for a vertical flow is the sum of three contributing factors: hydrostatic pressure gradient, friction pressure gradient, and acceleration pressure gradient.

Leklong et al [1] dealt with the dynamic responses of top end excitation of marine risers/pipes conveying internal fluid using a flexible link between undersea bore head and subsurface offshore platform. Chucheepsakul et al [2] developed a vectorial formulation of the mathematical model for stability analysis of flexible marine pipes conveying fluid. The model presented features high extensibility of the beam

60 Patrick S. Olayiwola et al.: Dynamics of a Flexible Offshore Gas Transmission Riser with Random Vortices

due to large axial strain and cross-sectional change due to Poisson effect among others.

Smith [3], Poettmann [4], Cullender and Brinkley [5], and other authors have in their investigations assumed the acceleration terms to be negligible, the flow to be steady and isothermal, and that no work is done by the gas in flow. Olunloyo et al [6] investigated the dynamics of offshore risers and pipeline systems with the sea environments defined by both the Laplace and Bernoulli’s Equations.

Now, for gas transmission through piping systems, researchers have mostly used the general form of the energy equation by applying the law of conservation of energy between two points on the fluid-conveying riser/pipeline system. (Poettmann & Carpenter [7], Cinedinst [8]). Also, Osheku et al [9-10] investigated the vibration of an offshore gas-conveying pipeline, as well as the analysis of flow induced acoustic waves in a vibrating offshore pipeline. In this paper, however, we are investigating the possibility of a stochastic vortex flow, caused by the acceleration terms that may occur, in addition to the pressure gradient gas transmission means. Thus, the effect of force of the vortex-induced vibrations on the riser is also examined.

2. Mechanics of the Ocean Environment The environment in which the beam conveying a single

phase compressible fluid is interacting comprises of the ocean water and ocean floor elastic mud. These aspects of the environment influence the dynamic and vibration responses of the riser under investigation. Firstly, the Euler equation of motion in an inviscid and incompressible fluid i.e. with constant density ρ , then u being the velocity vector, under the action of body force F is given as

1= − ∇ +

u Fρ

D PDt

(1)

and since the flow is assumed to be incompressible, the equation of continuity is given by

0∇ ⋅ =u (2)

where D Dt is a material derivative for the ocean kinematics. Now, one of the fundamental properties of a fluid flow is the vorticity ω that measures the local spin or rotation of individual fluid particles and is given as

= ∇ ×uω (3) On assuming that the flow is irrotational and with

incompressibility, also non-viscous, there exists the velocity potential ϕ and stream function ψ in our analysis.

2.1. Essential Fluid Mechanics

Given that the flow is irrotational, there exists a velocity potential and with the flow being incompressible, the continuity equation (2) is satisfied. Hence, the required equations governing the ocean environment in which the gas-conveying riser system is operating are given as follows;

2 0∇ =ϕ (4a)

and 2 0∇ =ψ (4b)

together with the Bernoulli’s equation of the form 2

02

∂∇ + + + = ∂

ϕρWP vgz

t (4c)

2.1.1. Ocean Water Response

For Eqns. (4), we consider superposition of the basic flow solutions for (i) the uniform flow, (ii) the source and sink flow, and (iii) the free vortex flow, as adequately describing the flow around the vertical cylinder, thus resulting in the following solutions;

2

0 21 cos

= −

θeR

w

rU UR

(5a)

and

2

0 21 sin2

Γ= − + −

θ θ

πe

ww

rU URR

(5b)

from the Bernoulli equation (4c), considering a steady potential flow we deduce that

20

2

∇ + + = ρ

W

w

P v gz (6a)

Taking the vertical component along the potential flow gives

1 0∂ ∂+ + =

∂ ∂ρW

W

P vv gz z

(6b)

neglecting the nonlinear term, Eqn. (6b) becomes

∂= −

∂ρW

WP gz

(7)

2.2. Mud-Soil Pressure Gradient

It is assumed that the ocean floor obeys the Darcy’s law, having the geo-mechanical properties or characteristics of the ocean- mud sediment gradient modeled as

∂=

∂

µs p ss

c

C UPz k

(8)

2.3. Gas Flow Mechanics In the following analysis, the compressible fluids under

investigation are perfect gases or dry vapours which behave like perfect gases. The choice of variables that is preferred is based on an appreciable density changes occurring across the transmitting ends. Thus, instead of using the volume flow

International Journal of Mechanics and Applications 2014, 4(3): 59-79 61

rate Q , the mass flow rate .

m is used; likewise for the

head change h , the isentropic stagnation enthalpy change

0h is employed. The choice of this last variable is a

significant one for in an ideal and adiabatic process, 0h is equal to the work done by unit mass of fluid. Since heat transfer from the manifolds is, in general, of negligible magnitude compared with the flux of energy through the compressing machine, temperature on its own may be safely excluded as a fluid variable. However, temperature is an easily observable characteristic and, for a perfect gas, can be easily introduced at the last parts by means of the equation of state.

2.3.1. Gas Dynamics

For a gas, the continuity equation is given as,

( )0∂

+ ∇ ⋅ =∂

Vρ ρt

(9)

and the energy equation as, 2

2

- P2 k "'

− ∇ ⋅ ⋅∇ + + = + ∇ − ∇ ⋅ +

V V

qρ

r

PD V gz uDt T q

(10)

then for the state equation of gas as = ρP RT (11)

where

∂ ∂ ∂ ∂= + + + ∂ ∂ ∂ ∂

x y zD V V VDt t x y z

(12)

However, heat radiation term qr is negligible; no internal heating "'q is involved, thus in the direction of flow, the energy equation (10) is rewritten as

2

2∂ ∂ ∂ ∂ + + + + ∂ ∂ ∂ ∂ ∂ ∂ = − − + ∇ ∂ ∂

ρ V V u uV gV Vt z t z

V PP V k Tz z

(13)

but

;∂∂= =

∂ ∂ γpv

vcc Tu c

z z (14)

where ,v pc c

are specific heats at constant volume and pressure respectively. Now Eqn. (11) results in

2 2 2

2 2 2

1 ;

1

∂ ∂ ∂ = − ∂ ∂ ∂ ∂ ∂ ∂

= − ∂ ∂ ∂

ρρ ρ

ρρ ρ

T P Tz R z z

T P TRz z z

(15)

Substituting Eqns. (14) and (15) into Eqn. (13) and with ∂ ∂ = ∂ ∂V V z V t , we write as follows;

2 2

2 2

2 2

2 1

0

∂+ + ∂

∂ + + ∂

∂ ∂ ∂− + − =

∂ ∂ ∂

ρρ

ρ ρρ ρ

v

v

P VV gVV t

c PVR z

kTV k Pc VTz Rz z

(16)

As the change in the internal fluid density assumes zero, then Eqn. (16) becomes

2

2

2 2

2 1 0

∂+ + ∂

∂ ∂ + + − = ∂ ∂

ρ ρρ

ρv

P VV gVV t

c P k PVR z R z

(17)

the acceleration term ∂ ∂V t is assumed to be negligible since the flow velocity is uniform, and then Eqn. (17) will reduce to

2

22 1 0

2∂ ∂ + + − = ∂ ∂

ρρ

vc P k PgV VR z R z

(18)

2.3.2. Gas Pressure and Enthalpy

We assume linear pressure drops as the gas flows through the riser, it is clear that, since the cross-sectional area of the beam is uniform, the volume increases along the height of the riser. Thus, the volume per unit mass v , may become

=v Az (19) and the pressure and enthalpy drops, can be given as; for the linearly decaying pressure,

( )1

−=

L zP P

L (20)

then enthalpy h of the flow which also drops with distance along the flow direction as

= +h u pv (21a)

and using Eqns. (19) and (20), we get

1 1 = + −

z zh u P AL L

(21b)

For which the pressure and the enthalpy gradients are 2

12; 0∂ ∂

= − =∂ ∂

PP Pz L z

(22)

and

1 21∂ = − ∂

P Ah zz L L

(23)


Substituting Eqn. (22) into Eqn. (18), gives

11 03

+= ∀ ≠ +

γργ

P gL V (24)

3. Governing Differential Equation (GDE) of the Vibrating Riser

riser

Figure 1. Model of an Offshore Gas-Conveying Riser

As shown in Figure (1) above, the structure is protruding from the ground (and connected to a flowline buried in the soil at the base of the ocean) to the surface of the ocean water. The soil rigidity, the flexibility and stringency of the ocean water forces and those of the gas dynamics on the riser are accounted for in the energy balance on the system. Thus, using Euler-Bernoulli beam theory for an initially bent riser as in Leklong et al [1, we write

2* 2

2

2

2

12

∂ ∂= + +

∂∂ ∂ − ∂∂ = − + ∂ ∂ + − ∂

κ

ε

w kk w uss

u kwsu kw

s w kus

(25a)

Meanwhile, for the purpose of this work, it is assumed that the riser is initially straight, hence 0=k , thus the strain energy becomes

( )

( ) ( )( ) ( )

* *

2 2

2 2

" "

' ' ' '

' '

= +

= + + +

∫

∫

δ κ δκ εδε

δ

δ δ

δ

U EI EA dz

EI w w

dzu u u uEA

w w

(25b)

for the kinetic energy

( )

( )

. . . .

. .

.

.

.

.

sin

sin

cos

cos

+ − +

+ + +

−

= × −

+ + +

× +

∫

ρ ρ ρ δ δ

θ δ θ

θ

δδ θ

ρθ

δ θ

p g W

p g W

g

w w u u

I I I

V w

KV w

V u

V u

(25c)

for the work done on the beam – these include, works due to the gas and ocean water hydrostatic pressure gradient and top tension, as described in Fig. 2(a) and (b), the gas enthalpy gradient, and inertia and buoyant forces, thus noting that

0∂ ∂ = ∂ ∂ =e iA z A z we write as follows;

( ) ( )

( )

( )( )

( )( )

0

2

sin ' ' cos ' '

' '

' '

' ' '

/

/

' ' ' '

−

∂ ∂ + − ∂ ∂ ∂ ∂ + − ∂ ∂ ∂ + += ∂ Λ + + + − × +

φ δ φ δ

δ

δ

δ δδ

µ

βρ

δ δ

t t

gWeW i

gses i

s v

viscous

W w

buoyant

T w w T u u

PPA A w wz z

PPA A w wz z

h w w u uW zK k U

F L

g L T T

F L

w w u u

∫ dz (25d)

where

( )2 ; 2 ;2

= − Λ =

= Λ

π ππ

eW e i i

es e

A r L A r LA r

(26)

Platform to storage, etc.

𝑧𝑧 = 0 Ocean top surface

Soil surface with stiffness cconstant 𝐾𝐾𝑆𝑆

Gas flow line (Reservoir) 𝑉𝑉1

𝐾𝐾𝑊𝑊 − Domain stiffness

𝑧𝑧 = −ℎ

Riser

𝜇𝜇 𝜔𝜔

𝜌𝜌𝑊𝑊

𝜇𝜇𝑊𝑊

Ocean rigid bottom surface


Figure 2. Riser elements subjected to pressure force and the top tension resolution

On applying Hamilton’s principle i.e.,

( )0

0− − = ∫ δ δ δT

U K W dt (27)

and substituting Eqns. (25) into Eqn. (27), then including the excitation force ( )F t , we deduce the governing differential as follows;

( )( )

( )

( )( )

2

0

2

""

sin ' '

2 ' ' " ( )

2

p g W

g t W eW s es

g i v

W w

p g W

EIw w

V T P A P A

P A h k U w F t

g L T T

g k w

ρ ρ ρ

ρ φ

µ

βρ

ρ ρ ρ

•• + + +

− − + + + + + = + −

+ + − +

(28)

where the term 2 2••

=w d w dt gives the material acceleration of the riser’s deflection. Hence, we write as follows

( )

( )

( )

( )( )

4 2

4 2

2

2

20

2

sin

' '( )

2 ' '

2

∂+ + +

∂ − − + ∂ + =

+ + + ∂ + − + + − +

ρ ρ ρ

ρ φ

µ

βρ

ρ ρ ρ

p g W

g t

W eW s es

g i v

W w

p g W

w d wEIz dtV T

P A P A w F tP A h k U z

g L T T

g k w

(29)

With 2 2d w dt being a material derivative, we have

2 2

2 2∂

=∂

ζ wd f wdt z

(30)

Therefore, Eqn. (29) becomes

( )

( )

( )

( )( )

4 2

4 2

2 2

2

2

0

2

1

sin' ' ( )

2 ' '

2

+ ∂ ∂+ +∂ ∂

+ + + − ∂ + − + = ∂ + + + + −

+ + − +

ψ

ρ ρ

ρ

ρ ρ δ δ ζ

φ

µ

βρ

ρ ρ ρ

p g

W

g p g W

t

W eW s es

g i v

W w

p g W

w wEIz t

V

TwP A P A F t

zP A h k U

g L T T

g k w

(31)

where

( )

( )( )

2 2 2 2 2 21 1 2 2

23 4 2

21 2 1 2 2

1 1 2 2

3 4

+ + + = + + + × +

ψ

λ β λ β

β β

ζ λ λ β β

λ β λ β

β β

wx wy

wz s

wx wye

wx wy

wz s

U U

U ULU Ur

U U

U U

ii AP

ee AP

TT

(a) Internal and external

pressure forces

( )−e e i iP A P A '

(b) Top tension

ϕ

ϕsintT

ϕcostT


( )( )

2 2 2 2 25 6 2

2 2 23 7 5 6 2

5 7 6 7

5 61 2 1

7

5 62 3 2

7

3 3 4

5 6 7

4

+ + + + + +

+ + + + + + + + × + +

β ω β ω

λ β ω β β ω ωπ

β β ω ω β β ω ω

β ω β ωλ λ β

β ω

β ω β ωλ λ β

β ω

λ β β

β ω β ω β ω

w s

g w s

w g g s

wx w wx s

wx g

wy w wy s

wy g

wz s

r w s g

pLp

p

pU UU

pU U

U

U U

p

2

2

π e

Lr

(32a)

this reduces to

( )

2 2 2 2 2 21 1 2 2

1 2 1 222 2

0 1 1 2 22

3 4

3 4

2 2 2 2 22 22 5 630 2

5 6

22 22 37 2

4

sin

4

R

R

Re

wz s

wz s

w s

w s

g

c cc c

LU c cr

c c

c c

pLp

VL

θ

θ

ψ θ

λ β λ βλ λ β β

ζ λ β λ ββ β

β β

β ω β ωλω

π β β ω ω

φλβ

π

+

+ = + + + + × +

+ + +

+

2

ir

( )( )

( )

( )

21 2 1 2 3 2

0 03 3 4

5 6

21 2 1 2 3 2

0 73 3 4

2 23

0 7 5 62

2

2

4

sin

+ + + +

× +

+ + + +

+ +

×

θ

θ

λ λ β λ λ βω

λ β βπ

β ω β ω

λ λ β λ λ ββ

λ β βπ

λω β β ω β ω

π

φ

R

wz se

w s

R

wz se

w s

g

i

c cLUc cr

p

c cLUc cr

L p

Vr

(32b)

which is further simplified to

( )

2 2 2 2 2 21 1 2 2 1 2 1 2

2 2 21 1 1 2 2

3 43 4

+ +

= + + + + +

θ θ

ψ θ

λ β λ β λ λ β βζ α λ β λ β

β ββ β

R R

R Rwz s

wz s

c c c cN c c

c cc c

( )2 2 2 2 2 2 2 22 3 5 6 5 6+ + +α λ β ω β ω β β ω ωw s w sSr p p

( ) ( )

( )( )

1 2 1 2 3 21 2 5 6

3 3 4

1 2 1 2 3 23 7

3 3 4

22 3 7 3 5 6

2 2 2 27 3 4

sin

sin

+ + + + +

+ + ++ + +

+

θ

θ

λ λ β λ λ βα α β ω β ω

λ β β

λ λ β λ λ βα β

λ β β φ

α α β λ β ω β ω

β λ α φ

RR w s

wz s

RR

wz s g

w s

g

c cN Sr p

c c

c cN

c c V

Sr p

V

(33)

where 2

20 0

2

20 0

1 cos ;

1 sin ;2

= = − =

Γ

= = − + −

θθ

ωθ ω

ω

θπ

e wRR w

W

e

wW

rUcU R

U rcU R UR

( )

00

20 1

32 3

sin; ;Re (2 ) ;

2 ; 2 ;

; 2 ;

= = =

= =

= =

θ

θ

φωω ω ρ µ

ω

ω π α µ ρ

α α π τ

gss g e

i

e e

e e i

VU r

r

Sr r U L r

LU r L r r

( )4 2 2 24 4=α π τiL r (34)

and 0, 00 1, 0p

p P zp = =⟨ ≤ ∂ ∂ =

Γ= , then 1 2 3, ,λ λ λ are the

coordinate scaling factors, Sc is the Schmidt number, Re is the Reynolds number, Sr is the Strouhal number, p is the probability of ocean water vortex-formation due to the thermohaline circulation that is driven by the variations in water density imposed at the sea surface by exchange of ocean heat and water with the atmosphere, including a buoyancy exchange or a form of circulation possibly occurring due to the wind-driven effect which is forced by wind stress on the sea surface, and includes a momentum exchange, and 1 2 7, ,...,β β β are direction parameters.

4. Solution Analysis On substituting Eqns. (7), (8), (22), (23) and (24) into Eqn.

(31) the result is

( )( )

( )

( )

4 2

4 2

2 2

2

20

2

1

1

sin ' '2 ' '

∂ ∂+ + +

∂ ∂ + + + − − + ∂

+ ∂+ + +

+ −

ψ

ρ δ δ

ρ δ ρ δ δ ζ

φ

µ

βρ

p g W

p g p g W

t W eW s es

g i v

W w

w wEIz t

V

T P A P A wzP A h k U

g L T T


( )( )1 2 ( )+ + − + =ρ δ δp g W g k w F t (35)

where

1

11

' ; ' ;

2 1' 1 ; ;3

= − = −

+ == − = +

ρ

γργ

g W W

i

P P L P g

P A zh P gLL L

and Eqn. (35) reduces to a dimensionless form as follows

( )

( )

4 2

4 2

22

1 2

1( )

∂ ∂+ + +

∂ ∂ = ∂

+ Ψ − + ∂

δ δg W

p e

w wz t F t

wf z K wz

(36)

where

( ) ( )( )

( )

4

2 212

2 1 3

; ; ; ;

1 2 ;

1;

= = = =

= + − + + + + − Ψ = − − + + +

ψ

δ ρ ρ δ ρ ρ

δ δ

δ δ δ ζ

g g p W W p

e g W

g g g W t

R s p R w

w w L z z L

K L EI g k

V b T

b N c f b N a Ga Gr

( )2 2 31 1 1; ;= = =τ p ib L f P A L EI F FL EI

( ) ( )( )

( )

( ) ( )

2 23

22

4

32

2

20

2

; sin ;

; ;

2 ;

2; ;

2 2;

= =

= =

= − Λ =

−=

= =

µ φ

µ

π τ ρ

βρµ

ρ ρµ µ

t

s s s p es s

W p

w e w

e eR

b L EI T L EI

b L a EI a C A k

a L re L EI

g r T TGr

U r r gN Ga

(37)

and

( )

( )

2 2 2 2 2 21 1 2 2

1 2 1 22 2 2 2

1 1 1 2 2

3 4

3 4

2 2 2 2 2 2 2 22 3 5 6 5 6

+

+ = + + + + × +

+ + +

θ

θ

ψ θ

λ β λ βλ λ β β

ζ α λ β λ ββ β

β β

α λ β ω β ω β β ω ω

R

R

R R

wz s

wz s

w s w s

c cc c

N Sc c cc c

c c

Sr p p

( )( )

( )( )

1 2 1 2 3 21 2

3 3 4

5 6

1 2 1 2 3 23 7

3 3 4

22 3 7 3 5 6

sin

+ + + +

× +

+ + ++ + +

θ

θ

λ λ β λ λ βα α

λ β β

β ω β ω

λ λ β λ λ βα β

λ β β φ

α α β λ β ω β ω

RR

wz s

w s

RR

wz s g

w s

c cN ScSr

c c

p

c cN Sc

c c V

Sr p

2 2 2 27 3 4 sin+β λ α φgV (38)

where ( )21= Ψ −B pC f z

represents the bending

coefficient. Eqn. (36) is the governing partial differential equation of the gas transmitting flexible riser idealized as an Euler-Bernoulli vertical beam, having part of its height measured Λ and buried in the soil under the ocean-bed, and the other part passing through the ocean to the surface, where the top end tension tT is significant.

4.1. Vibration Analysis of the Vibrating Riser

We now apply the even form of the Fourier transforms as follows;

( )

( )( )

( ) ( )

( )

4 44

4 3 3 10

22 2 1

02

2 22

2 2 2 10

,

cos

, cos

1 ,

2 cos

F

Fzz

FF

F

Fzz

n w sd wdz n w n w n z

d w n w s n w n zdz

n n w sd wzdz w n w n z

π λ

π π π

π λ π π

π π λ

π π

ℑ = + − ℑ = − − + ℑ = + −

(39)

also, using the Laplace transforms, we deduce the following;

( )2

22 , (0) (0)

ℑ = − −

λLs

d w s w s sw wdt

(40)

Solving Eqn. (38) by substituting Eqns. (39) and (42) accordingly, and writing the normal function for the problem viz a viz

( ) ( )( )( )( )

( )( ) ( )

( )

4 4 3 3

2

2 2 21

2 2

, (0, ) (0, ) 1

, ( ,0) ( ,0) 10

1 ,

" 2 1 ( , )

F F

Lg W

Fp

Fe

zzn w s n w s n w s

s w s sw w

n n f w s

w n w K w s

π λ π π

λ λ λ δ δ

π π λ

π λ

+ − + − − + +

= − Ψ − +

+ − +

(41)


With zero initial conditions, Eqn. (41) becomes

( )( )( )

2 4 4

2 2 21

10

1

+ + + = − Ψ − + +

δ δ π

π π

g W F

p e

s nw

n n f K (42)

Since 2 2 0+ Ω =s , and 0≠Fw , then Eqn. (42) is solved to give

( )( )( )

4 4 2 2 211

1

− Ψ − + +Ω = ±

+ +

π π π

δ δp e

g W

n n n f K (43)

where 2Ψ eand K are as defined above.

4.2. Dynamic Analysis of the Vibrating Riser

The system’s response for a supported offshore gas-conveying column is given, by applying Eqn. (39) and (40) on the governing differential equation (38) to yield

( )2 2+ Ω =−

τ

ω

sF Fes w

s i (44)

which further gives

( )( )( )2 2

, =− + Ω

τλ

ω

sF Few s

s i s (45)

and therefore becomes

( ) ( )2 2 2 2cos sin,

FF F

F F

i t t i tew t Fω ωλω ω

− Ω Ω − Ω = + Ω + Ω − Ω

(46)

Thus, the solution to the boundary value problem of an offshore gas conveying riser is given as

( )( )

2 2

1 2 2cos sin,

sin

−

∞

=

Ω + Ω Ω − Ω= + Ω − Ω ×

∑

ω

ωω

ω

π

Fi t

FF

Fn

F

e

t i tw z t F

n z

(47)

or

( )

( )

2 23 3

2 2

2 3

,cos sin

6 4 12

−

Ω + = Ω Ω − Ω +

Ω − Ω

× − +

ω

ωπ

ωω

Fi t

FF

F

F

e

w z t n Ft i t

z z z

(48)

5. Analysis of Results and Conclusions In this paper, an offshore riser which transports highly

pressurized gas molecules is investigated. The interaction between both the external and internal fluid flows on the riser occurs as a result of the nonlinear resonance of the cylindrical body due to the vortex shedding lock-in and possibly the acceleration of the internal flow and high pressurization. At the period when vortices are formed and shed inside the ocean, the structure’s vibration is synchronized with it, for which the motion amplifies over time, and hence results in amplified vortex induced vibration (AVIV). On the other hand, the rotational portion of internal flow of the highly pressurized gas gratifies the system’s vibration, thereby elevating stresses on the riser and its end connections causing a tendency to pull-off alignment. The top tension of the riser as it is inclined to the vertical at a very small angle φ causes an internal flow rotation as the flow velocity gV

creates a lateral component sinφgV . But

since φ is small, then sin →φ φ , hence as presented in the following cases:

Case 1: For the outward radial and upward vertical flows, and no circulatory flows, that is

1 2 3 4 5 6 71; 0= = = = = = =β β β β β β β and that

1 2 3 1= = =λ λ λ , then

( )

2 2

2 2 21 1

+ +

= + + + + +

θ θ

ψ θζ αR R

R Rwz s

wz s

c c c cN c c

c cc c

(49)

Case 2: For the outward radial and circumferential flows are clockwise, and upward vertical flow i.e. all are positive:

1 2 3 4 5 6 7 1= = = = = = =β β β β β β β and that

1 2 3 1= = =λ λ λ , then

( )

( )( )

( )

2 2

2 2 22 1

2 2 2 2 22

1 2

3

2 3

2 24

R R

R Rwz s

wz s

w s w s

RR w s

wz s

RR

wz s g

w s

g

c c c cN c c

c cc c

Sr p p

c cN Sr p

c c

c cN

c c VSr p

V

θ θ

ψ θ

θ

θ

ζ α

α ω ω ω ω

α α ω ω

αφ

α α ω ω

α φ

+ +

= + + + + +

+ + +

+ + + + +

+ + ++ + +

+

(50)


Case 3: For the inward radial and circumferential flows are anti-clockwise, and downward vertical flow i.e. all are negative: 1 2 3 4 5 1= = = = = −β β β β β 6 7 1= = −β β

and that 1 2 3 1= = =λ λ λ , then

( )

( )

2 2

2 2 23 1

2 2 2 2 22

R R

R Rwz s

wz s

w s w s

c c c cN c c

c cc c

Sr p p

θ θ

ψ θζ α

α ω ω ω ω

+ +

= + + + + +

+ + +

( )

( )( )

1 2

3

2 3

RR w s

wz s

R R wz sg

w s

c cN Sr p

c c

N c c c cV

Sr p

θ

θ

α α ω ω

αφ

α α ω ω

+ + + + +

+ + +− − +

(51)

Case 4: For the inward radial and upward vertical flows, internal circumferential flow is clockwise, ocean and mud circumferential flows are anti-clockwise, thus

1 5 6 1;= = = −β β β 2 3 4 7 1= = = =β β β β and taking

1 2 3 1= = =λ λ λ , then

( )( )

( )( )( )

( )( )

2 22 2 2

4 1

2 2 2 2 22

1 2

3 2 24

2 3

2 24

R RR

R wz s wz s

w s w s

R R wz s w s

R R wz sg g

w s

g

c c c cN

c c c c c c

Sr p p

N Sr c c c c p

N c c c cV V

Sr p

V

θ θψ

θ

θ

θ

ζ α

α ω ω ω ω

α α ω ω

αφ α φ

α α ω ω

α φ

+ −= − − − − +

+ + +

+ − − − +

− − −− + + +

+

(52)

For cases 1-5, the above figures demonstrate both the natural frequencies and response profiles for the system of an offshore gas-conveying riser. As observed in Figs. 3-8, monotonically decreasing and increasing profiles of the natural frequencies against increasing internal flow velocities are depicted. It is noticed that the decreasing trend continues until the critical velocities are reached, after which the frequencies turn to increase exponentially. The same profile patterns are noticeable when the natural frequencies are plotted against the Reynolds number of systems environment. The monotonic decreases begin at the subsonic regions and traverse to the critical value of the Reynolds number and further increases in values even to the supersonic regions gives increasing values of the frequencies. (Figs. 8-11).

Case 1: 1 2 3 4 1= = = =β β β β 5 6 7 0= = =β β β

1 2 3 1= = =λ λ λ

Figure 3. Graph of Ω versus gV

0 5 10 150

10

20

30

40

50

60

70

80

90

100

Internal flow velocity

Nat

ural

freq

uenc

y

n=1n=2n=3n=4)

L=300m; Uo=0.2m/s


Case 2: 1 2 3 4 1;= = = =β β β β

5 6 7 1;= = =β β β 1 2 3 1= = =λ λ λ


Case 3: 1 2 3 4 1= = = = −β β β β

5 6 7 1;= = = −β β β

1 2 3 1= = =λ λ λ


0 5 10 150

20

40

60

80

100


Nat

ural

freq

uenc

y

n=1n=2n=3n=4)

L=300m; Uo=0.2m/s; Pr=0.75

0 5 10 150

20

40

60

80

100


Nat

ural

freq

uenc

y

n=1n=2n=3n=4)

L=300m; Uo=0.2m/s; Pr=0.75


Case 3: 1 2 3 4 1= = = = −β β β β ;

5 6 7 1= = = −β β β

1 2 3 1= = =λ λ λ


Case 4: 1 5 6 1;= = = −β β β

2 3 4 7 1= = = =β β β β ;

1 2 3 1= = =λ λ λ


0 2 4 6 8 10 12 14 16 180

200

400

600

800

1000

1200

1400

Normalized V

Nat

ural

fre

quen

cy

n=1n=2n=3n=4)

L=1000m; Uo=10m/s

0 5 10 150

20

40

60

80

100


Nat

ural

freq

uenc

y

n=1n=2n=3n=4)

L=300m; Uo=0.2m/s; Pr=0.75


Case 1: 1 2 3 4 1;= = = =β β β β

5 6 7 0= = =β β β

1 2 3 1= = =λ λ λ

Figure 8. Graph of Ω versus RN

Case 3: 1 2 3 4 1= = = = −β β β β ;

5 6 7 1= = = −β β β

1 2 3 1= = =λ λ λ


0 2 4 6 8 10 12 14 16 18

x 106

0

1

2

3

4

5

6

7

8

9

Reynolds Number

Nat

ural

freq

uenc

y

V=5m/sV=7m/sV=10m/sV=15m/s)

L=600m; Pr=0.37

0 2 4 6 8 10 12 14 16 18

x 106

0

2

4

6

8

10

12

14

16

18

Reynolds Number

Natu

ral f

requ

ency


L=300m; Pr=0.37


Case 4: 1 5 6 1;= = = −β β β

2 3 4 7 1= = = =β β β β ;

1 2 3 1= = =λ λ λ


Case 2: 1 2 3 4 1= = = =β β β β ;

5 6 7 1= = =β β β

1 2 3 1= = =λ λ λ


0 2 4 6 8 10 12 14 16 18

x 106

0

5

10

15

20

25

30

35

Reynolds Number

Nat

ural

freq

uenc

y


L=300m; Pr=0.37

0 2 4 6 8 10 12 14 16 18

x 106

0

10

20

30

40

50

60

Reynolds Number

Natu

ral f

requ

ency


L=600m; Pr=0.37


As for the profiles of the natural frequencies against the Strouhal numbers, Figs. 12-14 depict these. At the subsonic values of the Reynolds number, increased value of the internal flow velocity causes significant effects on the nature of the frequency profiles. However, modest values of flow velocity demonstrate monotonically decreasing and increasing profile trends, as can be observed for other parameters.

Case 3: 1 2 3 4 1= = = = −β β β β ;

5 6 7 1= = = −β β β

1 2 3 1= = =λ λ λ

Figure 12. Graph of Ω versus Sr

Case 2: 1 2 3 4 1= = = =β β β β ;

5 6 7 1= = =β β β

1 2 3 1= = =λ λ λ


0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Strouhal Number

Nat

ural

freq

uenc

y


L=300m; Uo=2m/s; NR=150

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Strouhal Number

Natu

ral f

requ

ency


L=300m; Uo=2m/s; NR=150


Case 4: 1 5 6 1;= = = −β β β

2 3 4 7 1= = = =β β β β ;

1 2 3 1= = =λ λ λ


Figs. 15-18 describe the dynamic responses profiles for this system. Parabolic curves, as expected, are demonstrated and the boundaries of the system as constrained are satisfied. More so, the fact that at time 0=t , the system deflects, defines its initial configuration, which are shown in Figs. 19-22. The 3-D shapes of the response profiles also corroborate the parabolic characteristic of the responses of the system. (Figs. 23 - 24).

Case 1: 1 2 3 4 1;= = = =β β β β

5 6 7 0;= = =β β β

1 2 3 1= = =λ λ λ

Figure 15. Graph of W versus z

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

Strouhal Number

Natu

ral f

requ

ency


L=300m; Uo=2m/s; NR=150

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8x 10

-7

Depth of riser

Res

pons

e

n=1n=2n=3n=4)

L=300m; Uo=0.2m/s;


Case 3: 1 2 3 4 1= = = = −β β β β ;

5 6 7 1= = = −β β β ;

1 2 3 1= = =λ λ λ


Case4: 1 5 6 1;= = = −β β β

2 3 4 7 1= = = =β β β β ;

1 2 3 1= = =λ λ λ


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-8

-6

-4

-2

0

2

4x 10

-7

Depth of riser

Res

pons

e

n=1n=2n=3n=4)

L=300m; Uo=0.2m/s;Pr=0.75

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2.5

-2

-1.5

-1

-0.5

0

0.5x 10

-6

Depth of riser

Res

pons

e

n=1n=2n=3n=4)L=300m; Uo=0.2m/s;Pr=0.75


Case 4: 1 2 3 4 1= = = =β β β β ;

5 6 7 1= = =β β β ;

1 2 3 1= = =λ λ λ , and 0=p


Case 1: 1 2 3 4 1;= = = =β β β β

5 6 7 0= = =β β β ;

1 2 3 1= = =λ λ λ

Figure 19. Graph of W versus dimensionless time

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-8

-6

-4

-2

0

2

4x 10

-7

Depth of riser

Res

pons

e

n=1n=2n=3n=4)

L=300m; Uo=0.2m/s;Pr=0.75

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18-8

-6

-4

-2

0

2

4

6

8

10x 10

-6

t/τ

Res

pons

e

n=1n=2n=3n=4)

L=1000m; Vg=50m/s; Uo=10m/s


Case 2: 1 2 3 4 1= = = =β β β β ;

5 6 7 1= = =β β β ;

1 2 3 1= = =λ λ λ


Case 3: 1 2 3 4 1= = = = −β β β β ;

5 6 7 1= = = −β β β ;

1 2 3 1= = =λ λ λ


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14-8

-6

-4

-2

0

2

4

6x 10

-7

t/τ

Res

pons

e

n=1n=2n=3n=4)L=1000m; Vg=50m/s; Uo=10m/s

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18-8

-6

-4

-2

0

2

4

6

8x 10

-7

t/τ

Res

pons

e

n=1n=2n=3n=4)

L=1000m; Vg=50m/s; Uo=10m/s


Case 4: 1 5 6 1;= = = −β β β

2 3 4 7 1= = = =β β β β ;

1 2 3 1= = =λ λ λ


Figure 23. 3-D shape of the dynamic profile

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

-6

Normalized time

Res

pons

e

n=1n=2n=3n=4)

L=1000m; Vg=50m/s; Uo=0.2m/s

00.2

0.40.6

0.81

00.05

0.10.15

0.20

0.5

1

1.5

2

x 10-6

Ocean depthTime

Res

pons

e


Figure 24. 3-D shape of the dynamic profile

Nomenclature w the transverse deflection ω the angular velocity of the gas F body force Φ the velocity potential; Ψ the stream function;

WP the pressure of the ocean water;

ρW the density of water;

ρ p the density of the riser;

V the velocity vector; u the velocity vector of water flow; Ps mud-sediment pressure;

1P the gas pressure at the riser base;

gP

the gas pressure;

sA the surface area in contact with the sediment

µs bed sliding coefficient

µw the dynamic viscosity of the water µ the gas viscosity; Cp porosity coefficient or index;

sK bed’s specific permeability coefficient;

hP mean pressure/volume; F the actuating force;

, ϕpV V the gas velocities due to pressure gradient and circulation respectively;

V the (internal) gas flow velocity;

0V the mean gas flow velocity; EI the flexural rigidity of the pipe (riser);

,p gA A

pipe, riser x-sectional areas g acceleration due to gravity; t the time variable; Λ riser’s depth in the soil under the sea

wH hydrodynamic loading of the seawater from the free surface.

h enthalpy of the gas.

sk soil stiffness constant β coefficient of cubical expansion of the fluid;

the gas constant; the adiabatic gas constant;

Γ the strength of the vortex circulation;

er the external radius of the cylinder; , θRU U radial and circumferential velocities;

WU the stream velocity; u the internal energy; R the gas constant. ρ the gas density;

pc

gas specific heat capacity at const. pressure;

k the medium conductivity. φ the angle between top tension & vertical axis

tT the Top tension;

wT the pipe wall temperature;

the gas temperature;

the ocean uniform flow velocity.

the excitation force

the internal radius

00.2

0.40.6

0.81

00.01

0.020.03

0.040.05

0

0.5

1

1.5

2

x 10-5

Normalized riser lengthNormalized time

Res

pons

e

L=1000m; n=1;Vg=50m/s;Uo=10m/s

Rγ

T0U

Fir


external radius Sc the Schmidt number

RN the Reynolds number Sr the Strouhal number p the probability of vortex-formation Gr the Grash of number

eK the system’s apparent stiffness constant

REFERENCES [1] Leklong J, Chucheepsakul S and Kaewuunruen S (2008)

“Dynamic responses of Marine risers/pipes transporting fluid subject to top end excitations.”Proceedings of the Eighth (2008) ISOPE Pacific/Asia Offshore Mechanics Symposium.

[2] Chucheepsakul S, Huang T and Monprapussorn T (2001) “Stability analysis of extensible flexible marine pipes transporting fluid.” Proceedings of the Eleventh (2001) International Offshore and Polar Engineering Conference, Stavanger, Norway, June 17-22, 2001.

[3] Smith, R.V., “Determining Friction Factors for Measuring Productivity of Gas Wells.” Trans. AIME, 189:73 (1950).

[4] Poettmann, F.H., “The Calculation of Pressure Drop in the Flow of Natural Gas Through Pipe.” Trans. AIME 192-317 (1951).

[5] Cullender, M.H. and Brinkley, C.W., “Adaptation of the Relative Roughness Correlation of the Coefficient of Friction to the Flow of Natural Gas in Gas Well Castings.” Report presented to Railroad Commission of Texas Amarillo, Texas Nov. 9, 1950.

[6] Olunloyo, V. O. S., Osheku, C.A. and Oyediran, A.A. (2007) “Dynamic Response Interaction of Vibrating Offshore Pipeline on Moving Seabed” ASME Journal of Offshore Mechanics and Arctic Engineering, Volume 129, No. 2, pp. 107-119.

[7] Poettmann, F.H. and Carpenter, P.B., “The Multiphase Flow of Gas, Oil, and Water Through Vertical Flow Strings with Application to the Design of Gas Lift Installations.” Drilling and Production Practice (1952) 257.

[8] Cinedinst, W.P., “Flow Equations for Gas Considering Deviations from Ideal Gas Laws.” Oil and Gas J. 43(48):79 (1945).

[9] Osheku C.A, Olunloyo V.O.S and Olayiwola P.S (2010)“On the Mechanics of a subsea Gas Pipeline Vibration”; 29th ASME International Conference on Ocean, Offshore and Arctic Engineering, Paper No. OMAE2010-20066, Shanghai, China, June 6th- June11th 2010.

[10] Osheku, C. A., Olunloyo, V. O. S., Akano, T. T. (2010)" Analysis of Flow Induced Acoustic Waves in a Vibrating Offshore Pipeline"; Proceedings of the 29th ASME International Conference on Ocean, Offshore and Arctic Engineering, Paper No. OMAE2010-20060, Shanghai, China, June 6th- 11th, 2010.

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