Lecture 1-b Catenary Risers: Global Analysis

Lecture 1-b

Catenary Risers: Global Analysis

Dynamic Problem

equations and time scales

Celso P. Pesce

Professor of Mechanical Sciences

PhD in Ocean Engineering, MSc Marine Hydrodynamics, Naval Architect

[email protected]

LMO - Offshore Mechanics Laboratory

Escola Politécnica

University of São Paulo

Brazil

Catenary lines

junta

flexível

riser de aço

bóia

intermediária

poita

enrijecedores de

flexão

TLPFPSO

flutuadores

restritor de

curvatura

flexible joint

Steel catenary riser

Bending restrictors

Subsurface buoyFloaters

Bending stiffener

Lazy-wave riser

Weight anchor

Qv

Qw

bn

P’v

w

t=u

T

Mu

Mv

Mw

2

2

2

2

2

2

2

2

2

2

2

2

)(KQMMM

)(KQMMM

)(KMMM

)(fQTQ

)(fTQQ

)(fQQT

tIt

s

tIt

s

tIt

s

t

wmt

s

t

vmt

s

t

umt

s

w

wwwvuvvu

w

v

vvvwwuuw

v

u

uuuvwwv

u

wuvv

w

vwuw

v

uvwwv

General Kirshoff-Clebsh-Love Equations

(KCL equations)

Curved bars – large displacements, small deformations

The planar static problem

with:

)(cos

)(sin

scqf

scqf

nn

tt

(s)

y

xi

j

k

uv

. 0

0

0

Qds

dM

fds

dT

ds

dQ

fds

dQ

ds

dT

n

t

Static forces due to

current

)();( scsc nt

Perturbations around the

static equilibrium

configuration

2

2

2

2

cosΤQΤQ

sinQQ+TΤ

t

umqc

sds

d

s

t

umqc

sds

d

s

nnn

ttt

Hang-off

boundary-

layer

TDP

boundary-

layer (s,t)n t

yx

),()(),Q(

),()(),(Τ

),()(),(ε

),()(),(

tssQts

tssTts

tsests

tssts

eds

du

s

u

tssds

du

s

u

nt

tn

),())(1(

Kinematic relations

The planar dynamic problem

Oscillating hydrodynamic

forces

2

2

2

2

Q

cos

Q+T

sin

t

um

sds

d

s

qcds

dT

s

Q

t

um

sds

d

s

qcds

dQ

s

T

nn

n

tt

t

Separating static and dynamic parts:

Static equilibrium,

tangential direction

Static equilibrium,

normal direction

0

0

Linearized Dynamic Equations

Negleting second-order terms

Two dynamic equilibrium equations,

linearized around the static

equilibrium configurations

2

2

2

2

+

t

umT

s

ds

dQ

s

t

umQ

s

ds

dT

s

nn

tt

2

2

2

2

+

t

umT

sds

dQ

s

t

umQ

sds

dT

s

nn

tt

Kinematic relations

Constitutive equations

Negleting

bending stiffness

effects:

gSSq

uuqtsEAets

ds

du

s

u

eds

du

s

u

t

umT

sds

d

t

um

ds

dT

s

ifoaf

ntf

tn

nt

nn

tt

)cossin(),(),(

1

2

2

2

2

Linearized Dynamic Equations: neglecting bending stiffness

Two dynamic equilibrium equations,

linearized around the static

equilibrium configurations

)(sin)()(2

1)(

)(cos)()(2

1)(

2

2

sCsUsDUsc

sCsUsDUsc

Dan

Tat

Mean current hydrodynamic forces

nn

n

tt

t

Ut

u

Ut

u

wsinv

wcosv

),(),(),( tststs invisc

ttsU

t

umts

ts

nta

in

n

in

t

wcos)(),(

0),(

2

2

Oscillating hydrodynamic forces

Parcelas dinâmicas das forças devido à correnteza e às ondas:

Relative velocity

components to the

flow

Viscous parcels

Inertial parcels

nnn

naD

visc

n

tnt

taT

visc

t

s

usU

t

usUDCts

s

usU

t

usUDCts

wcos)(sin)(2vsinal2

1),(

wsin)(cos)(2vsinal2

1),(

Gravitational waves

velocity filed

s

cEI

ds

dEIQ

sEI

s

tscds

dEItsEI

2

2MQ

e

),(),(M

The dynamic equation in the normal direction may be written:

From the classic linear constituve equations:

The bending stiffness effect

2

2

2

2

cosT

Tt

umqc

sds

dEI

ssEI n

nn

Second-order

term

The nondimensional dynamic equation in the normal direction reads:

Defining:

L

T

EI

0

0

0T

q

Flexural lenght at TDP

Static curvature

at TDP

amm

Tc

tL

ct

00

0ˆ

Transversal wave celerity

due to geometric stiffness,

at TDP

2

2

02

2

22

ˆ

ˆcosˆˆˆ

ˆ

ˆ

ˆˆ

ˆˆˆ

ˆ

ˆ

t

uc

ssd

d

ss

nnn

The bending stiffness effect

Small length

parameter

Neglecting secon-order terms (bending stiffness):

Rigid risers (Steel): )10( 2O

Flexible risers: )10( 3 O

))(1(ˆ

ˆcosˆˆˆ

ˆ

ˆˆˆ 2

2

2

0

O

t

uc

s

nnn

The bending stiffness effect

Time scales

)()(

sT

EIs

Wave celerity associated to

geometric rigidity

Wave celerity associated to

axial stiffness

a

i

f

i

i

f

i

g

g

c

Lt

ct

c

Lt

ct

c

Lt

5

)(

)(

4

)(

)(

3

2

1

Local flexural length

Wave celerity associated to

bending stiffness

m

EAc

mm

EIc

mm

Tc

a

a

i

f

i

f

a

g

)(

2

)(

)(

)(

Model similarity and nondimensional group

[.

Number Symbol and definition Representation

Froude NumberDynamic motion in

waves

Reynolds NumberViscous forces

Strouhal NumberVortex shedding

frequency

Keulegan-

Carpenter Number

Inertial forces vs.

drag forces

Structural

Damping

Linear structural

damping

Reduced VelocityNormalized

velocity in VIV

Reduced Shedding

Frequency

Vortex shedding

normalized

frequency

Reduced MassRiser mass vs.

displaced mass

Added MassAdded mass vs.

riser mass

Bending Stiffness

Bending vs.

geometrical

stiffness

Axial Stiffness

Axial vs.

geometrical

stiffness

Soil Stiffness

Soil vs. bending

and geometrical

stiffness

gL

ArF

UDRe

U

Df stS

D

A2KC

cc

c

Df

UV

nr

rnn

ss V

Df

U

f

ff tt SS

Dm

mm

m

ma a

T

EI

LLK

f

f

1

T

EAKa

2T

EIkK s

s Rateiro et al

ISOPE2012

Summary

• The global dynamics is governed by the geometric stiffness or

catenary stiffness.

• Bending stiffness effetcs are importante at the extremities and

TDP, or for high-order vibration modes for which the mode

vibrattion length is of same order of the local flexural length.

• There are several time scales that govern the overall dynamics

of a riser.

• A large nondimensional parameters group govern the overall

riser dynamics

Acknowledgements

TPN

FLUID-STRUCTURE INTERACTION AND

OFFSHORE MECHANICS LABORATORY