Flow Induced Vibration

FLOW-INDUCED VIBRATION

This Page Intentionally Left Blank

PROCEEDINGS OF THE 7TH INTERNATIONAL CONFERENCE ON FLOW-INDUCED VIBRATION -FIV2000/LUCERNE/SWllZEKLANL)/ 19 - 22 J UNb ZOOU

Edited by

Samir Ziada

Thornas Staubli McMaster University, Hamilton, Ontario, Canada

Hochschule Technik + Architektur Luzern, Switzerland

A.A. BALKEMA/ ROTTERDAM/ BROOKFIELD / 2000

Photo cover: Vortex shedding in a normal triangle tube array. Courtesy of Sulzer Innotec, Winterthur, Switzerland

The texts of the various papers in this volume were set individually by typists under the supervision of each of the authors concerned.

Authorization to photocopy items for internal or personal use, or the internal or personal use of specific clients, is granted by A.A. Balkema, Rotterdam, provided that the base fee of per page is paid directly to Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, USA. For those organizations that have been granted a photocopy license by CCC, a separate system of payment has been arranged. The fee code for users of the Transactional Reporting Service is: 90 5809 129 5/00

per copy, plus

Published by A.A. Balkema, PO. Box 1675,3000 BR Rotterdam, Netherlands Fax: +3 1.10.413.5947; E-mail: [email protected]; Internet site: www.balkema.nl A.A. Balkema Publishers, Old Post Road, Brookfield, VT 05036-9704, USA Fax: 802.276.3837; E-mail: [email protected]

ISBN 90 5809 129 5 0 2000 A.A. Balkema, Rotterdam Printed in the Netherlands

Flow Induced Vibration, Ziada & Staubli (eds) 0 2000 Balkema, Rotterdam, ISBN 90 5809 129 5

Table of contents

Preface

Organizing Committee

Acknowledgements

XIII

xv XVI

Vortex-induced vibration A three-dimensional model for wave and vortex-induced vibration of deepwater riser pipes 3 LA. Ferrari Jr & FI W Bearman

Vortex induced vibrations measured in service in the Foinaven dynamic umbilical, 11 and comparision with predictions G.J. Lyons, J . K. Vandiver, C M. Larsen & G.TAshcombe

Predicting lock-in on drilling risers in sheared flows J. K. Vandiver

Vortex induced vibration of deep water risers R. H.J. Willden & J. M. R.Graham

Streamwise oscillations of a cylinder in a steady current 0. Cetiner & D. Rockwell

Free vibrations of two fix-supported elastic cylinders YZhou, R.M.CSo, Z.J.Wang & WJin

Vortex-induced vibration of a spring-supported cylinder in a narrow channel M. Yoshizawa, K. Sugiura & S Haga

Bifurcation and perturbation analysis of some vortex shedding models A? WMureithi, H. Kanki & T. Nakumura

21

29

37

45

53

61

On the mathematical modeling of vortex excited vibrations in bundled conductors P Hagedorn & ir: Hadulla An experimental investigation of some three-dimensional effects of pivoted circular cylinders L. G. Weiss & AA Szewczyk

69

75

A fluid force deduction technique for vibrating structures in cross-flow M. R. Gharib, A Leonard & M. Gharib

85

V

Dependence of flow-induced vibration parameters on spanwise trip-wires E S. Hover & M. S. Triantahllou

Fluid-structure interactions of two side-by-side circular cylinders R. M. C So, Y Liu & Y Zhou

Flow-visualization around a circular cylinder near to a plane wall S.J. Price, D. Sumner, J . G. Smith, K. Leong & M. R Puidoussis

Vibration of rectangular profiles in crossJEow 2D rectangular cross section aerodynamics in various incidence angle of wind M. Matsumoto, H. Shiruto & S. Honmu

Vortex-induced vibrations of an elongated rectangular beam I? Hkmon & I;: Santi

Stability analysis of an oscillating rectangular profile in cross-flow K. D. Kerenyi & 7: Stuubli

Numerical investigation of aerodynamic forces on rectangular cylinders and generic bridge deck sections X.Amundolese, I;: Bourquin & C. Cremonu

Aerodynamics of two edge girders for long-span cable-stayed bridges M. Mutsumoto, Y Daito & K.Aruki

Flutter characteristics of 2-box girders for super-long span bridges M. Mutsumoto, YTuniwuki, K.Abe & N Nukajimu

Oscillations of free shear layers and jets Influence on cavity shear flows by horizontal cover-plate with different leading-edge profiles C -H. Kuo & S.-H. Huang

Vortex dynamics and energy transport of self-sustained oscillating flow in the jet-cylinder interactions E B. Hsiuo, Y WChou & J. M. Huang

The interaction of a spanwise vortex with a rectangular plate J.M.Chen & H.-C.Chueh

Active stabilisation of a planar jet impinging on a flexible wedge S. Ziada

Effects of vibrations in a forced plane wall jet M. I;: Scibiliu

Vibration of hydraulic structures Flow-induced dynamic instability closely related to Folsorn Dam Tainter-Gate failure in California K.Anami & N Ishii

91

97

105

115

123

131

141

1 49

157

167

173

181

187

195

205

VI

How induced vibration of radial gate under small gate opening K. Ogihara, Y: Ueda & H. Ernori

Flow-induced vibration of Tormmbarry weir gates J. D. Hardwick, JAttari & J . Lewin

Single shear layer instabilities and vortex-induced excitation mechanisms I? Billeter Spectrum analysis evaluation of design modifications for improving butterfly valve emergency closing behaviour R.Angehrn

Applications of computational fluid dynamics Flow-induced vibration of an elastic circular cylinder at sub-critical Reynolds numbers Y: Liu, S. ?: Chan, R. M. C So & K. Lam

Numerical study of flow around circular cylinder S. T.Chun, K. Lam, R. M. C. So & R. C K. Leung

A numercal simulation of the response of a vortex-excited cylinder EGuilrnineuu & P Queutey

A numerical study of flow-induced cable vibration C-T.Hsu, M.-K. Kwan & C-CChang

Numerical study of the in-line oscillations of tube bundles S. Seitanis & PAnagnostopoulos

DNS-derived force distribution on flexible cylinders subject to VIV with shear inflow D. Lucor, C Evangelinos & G. E. Karniadakis

New approaches of wall models for large eddy simulation in complex geometries H. R. Barsarnian & Y:A. Hassan

A large eddy simulation of the turbulent forcing spectrum induced by axial flow on a rod M. M. Moreno, E.de Langre & P Le Quere

Application of an explicit coupling technique to open pool fluid-structure interaction G. D. Morandin & R.G. S a d

Coupled fluid-structure simulation: Instability problems under viscous flows N. Greffet & A. Cornbescure

Computational study on flow-induced vibration of high-speed train in tunnel M. Suzuki

Fluid-structure interaction: Axialjlows Solutions of unsteady confined viscous flows with separation regions for fluid-structure interaction problems D. Mateescu & D. Venditi

213

219

225

233

241

249

257

265

273

28 1

289

295

303

31 1

319

327

VII

Non-linear vibrations of circular cylinbcal shells with flow M.Arnabili, l? Pellicano & M. I? Paz'doussis

335

Dynamic stability of fluid-conveying cylindrical shells using a hybrid finite element approach 343 A.A. Lakis, M. I? Paz'doussis & C. Dupuis

Local and global instability of fluid-conveying cantilever pipes 0. Doare' & E.de Langre

349

Principle of equal presence in the problem of the dynamic interaction of a cantilevered pipe conveying fluid with elasto-viscous medium YA. Dzhupanov

355

Conservation laws in the dynamics of pipes conveying high-speed fluids YM.Vassilev, EA. Djondjorov & ??A. Dzhupanov

363

Influence of the boundary conditions in a friction based model of fluid-elastic instability in axial flow G. Porcher

37 1

Numerical investigation of the interaction between a compliant coating and an unstable 379 boundary-lay er 0. Wiplier & U Ehrenstein

The convective and absolute instabilities of flow over compliant surfaces K. SYeo, H.Z Zhao & B. C. Khoo

Fluid-structure interaction: Biomedical applications 3D Lagrangian model for capsule and Stokes flow interaction 0. Le Maitre, I? Navaro & S. Huberson

387

395

A new model for the vibration of an initially-stretched tube conveying pulsatile fluid flow 403 I! L. Zhang, D. G.Gorman & J. M. Reese

Panel flutter in a Poiseuille flow L. Huang

Aeroelastic model of vocal-fold vibration J HordCek & J. G.Svec

Interaction of cylindrical and spherical bodies in flowing ideal liquid ICKubenko, L. Kruk & YDzyuba

41 1

419

427

Aeroelasticity Nonlinear aeroelastic response of a two-degree-of-freedom airfoil oscillating in dynamic stall S.J. Price & G. Fragiskatos

437

Nonlinear stability of flutter-type vibration in wind J Ndprstek

Solution of flutter problems at Pilatus - Analysis, tests and verification A. Vollan

445

455

Vlll

A numerical study of forced vibrations of turbomachine blades 0. V Repetski

Wind-induced vibration of offshore platforms AA Petrov

Vibration of heat-exchanger tube bundles Characteristics of tube bundle vibrations in cross flow H. Tanaka, K. Tanaka & F: Shimizu

463

467

473

The effect of approach flow direction on the fluid-elastic instability of tubes in triangular tube arrays U. Mohr, K. Schroder & H. Gelbe

48 1

A case study of fluid-elastic instability of a large heat exchanger in a petrochemical process plant K. K. Botros, G. Price, D. May & C. Holding

489

The crossing frequency as a measure of heat exchanger support plate effectiveness M. K.Au-Yang

Analysis of a loosely supported beam under random excitations J. Knudsen, A. R. Massih & R.Gupta

Modelling turbulence response of heat exchanger tubes in loose supports M.A. Hassan, D. S. Weaver & M.A. Dokainish

Prediction of wear work rate and thickness loss in tube bundles under cross-flow by a probabilistic approach J . Charpentier & Th. Puyen

Effect of flow regime and void fraction on tube bundle vibration C. E. Taylor & M.J. Pettigrew

Some problems on the estimation of flow-induced vibration of a tube array subjected to two-phase flow 7: Nakamura, K. Hirotu & K. Tomomutsu

Modelling two-phase flow-excited fluid-elastic instability in tube arrays R Feenstra, D. S. Weaver & R. L. Judd

An experimental study on the fluid-elastic forces acting on a square tube bundle in two-phase cross flow - The effect of the vibration amplitude l? Inadu, K. Kawamuru, A. Yusuo & K.Yonedu

The effects of tube bundle geometry on vibration in two-phase cross-flow M.J. Pettigrew, C. E.Taylor & B. S. Kim

Flow-sound-structure interaction Non-linear flow/ acoustic interactions for two cylinders in cross flow J.A. Fitzpatrick

497

505

513

521

529

537

545

555

56 1

57 1

IX

Coupling effects in a two elbows piping system R Moussou, RVaugrante, M.Guivarch & D. Seligmann

579

The effect of sound on the heat transfer and the vortex-shedding from a heated circular 587 cylinder: Acoustical vibrations directed along its axis S. Okamoto

Predicting interior wind noise using a virtual wind tunnel G S. Strumolo

Interaction of a spherical shell with an acoustical medium I. Zolotarev

Optimisation of the location of a silencer in a hydraulic circuit A. Bihhadi & K.A. Edge

On noise radiated from a flow-induced vibrating elastic blade R. C K. Leung & R. M. C. So

Prediction model for broadband noise in bends J. WM.Gijrath, B.T.Verhaar & J. C. Bruggeman

Acoustic resonance in the inlet scroll of a turbo-compressor S. Ziada, A. Oengoren & A.Voge1

595

60 1

607

615

623

629

Flow-induced acoustic resonance in plate heat exchangers: Comparisons to large heat exchangers E. Rodarte & N R. Miller

637

Non-intrusive condition monitoring through digital signal analysis of noise data - A decade of progress M. K. Au-Yang

645

Prediction of valve noise on the basis of data from scale model tests J. C. Bruggeman, R. R. Parchen, J. WM.Gijrath & HJ. Riezehos

Acoustic fatigue of a steam dump pipe system excited by valve noise H. Pastorel, S. Michaud & S. Ziada

655

66 1

Flow-induced pulsations in pipe systems with closed branches, impact of flow direction M. CA. M. Peters & E.van Bokhorst

669

Aeroacoustic response of diffusers and bends: Comparison of experiments with quasi-stationary incompressible models S. Dequand, L.van Lier, A. Hirschberg & J . Huijnen

Thermo-acoustic instabilities Fluid dynamic instabilities in a swirl stabilized burner and their effect on heat release fluctuations C 0. Paschereit, P Flohr, W Pollfie & M. Bockholts

Case studies of burner/ furnace systems sensitive to thennoacoustic oscillations EL. Eisinger & R. E. Sullivan

677

687

695

X

Thermoacoustic instability in an exhaust gas stack H.R.Gruf & A.Oengoren

Thermoacoustic instability as the cause of the longitudinal gas oscillations in a tubular heat exchanger A. Ni

Forced leading-edge vortex shedding by arrays of rectangular cylinders I. K. Nwagwe & C M. Coats

Vibration of turbomachines and piping systems Estimation of vibrational state of mistuned blade assemblies of gas turbine compressor rotor wheel A. P Zinkovskii

Effect of vaned diffuser on performance and onset of pressure instabilities in centrifugal compressor G. Petelu & R. Motriuk

Noise of bubbly flow through orifice 0. Moc.lizuki & Wurjito

A transieqt dynaiaic analysis of a piping system subjected to a LOCA phenomenon S. k? Potapov & HAndriumbololona

Suppression of flow-induced vibrations by a dynamic absorber with parametric excitation H. Ecker & A. Tondl

Flow-induced vibrations related to rotors An improved linear model for rotors subjected to dissipative annular flows M. Moreiru, J. Antunes & H. Pina

Simulation study on vibration behavior of fluid journal bearing with incompressible fluid K. Fujita, A. Shintani & K. Yoshioka

Lubrication to structure interactions in thin hydrodynamic films H. Springer

Fluid-dynamic forces acting on the rotating inner cylinder in a concentric annulus WG. Sim & J. H. Park

Flow-induced vibrations related to paper machines Flow-induced vibration of free edges of thin films I/: B. Chang & I? M. Moretti

Leakage flow induced flutter of highly flexible structures S. Kaneko, S. dnaka, H. Fujituni & ?: Watunube

Instability of a blade for paper coating J . M.Genevaux, VMorin, J. R Maume & S. Skali

703

71 1

719

727

733

743

749

757

767

777

785

79 1

80 1

81 1

819

XI

leakage flow-induced vibration Vibrational behavior of a rigid body supported by a damper-spring system in a narrow passage with fluid K. Fujita, A. Shintani & TYoshino

Mechanisms of leakage-flow-induced vibrations - Single-degree-of-freedom and continuous systems F: Inada & S Hayama

829

837

Author index 845


Preface

This conference is the seventh in the series of International Conferences on Flow Induced Vibration, the first six of which were held in England. The change in venue symbolizes the conferences healthy growth. Since the first Keswick conference in 1973, with its focus on the needs of the nuclear industries, the scope of papers has become successively wider. These proceedings include more than 100 papers submitted from 25 countries and presented in 22 technical sessions within the Seventh International Conference on Flow Induced Vibration. A truly international event.

Flow induced vibrations and noise continue to cause problems in a wide range of engineering applications ranging from civil engineering and marine structures to power generation and chemical processing. This breadth of applications is exemplified by the major headings of the topics covered, whch include: - Vortex induced vibration; - CFD applications to fluid-structure interaction; - Advances in fluid-structure interaction theory; - Vibration of heat-exchanger tube bundles; - Flow- sound- struc ture interaction; - Oscillations of free shear layers and jets; - Thermo-acoustic instabilities; - Aeroelastici ty ; - Vibration of hydraulic structures; - Vibration of turbomachines and piping systems; - Flow induced vibration related to rotors and paper machines; - Leakage flow induced vibration; - Applications of fluid-structure interaction theory to biomedical engineering. The papers in these proceedings constitute a balanced mixture between papers from academia

and papers from industry. Thus, the conference brings together those working on the mechanisms and causes of vib,ntion with those in industry who are faced with either avoiding or solving flow induced vibration and noise problems.

The organizing committee is grateful to all the sponsors for their generosity. In particular, many thanks to SNF (Swiss National Funds) for supporting several researchers from Eastern European Countries enabling them to attend the conference and present the results of their research.

Xlll

We should like to thank the authors for providing papers of high quality and specifically for addressing such a wide spectrum of problems caused by fluid-structure interaction and flow-acoustic coupling. Sincere thanks are also due to all members of the organizing committee for their valuable advice and their support in bringing these papers together. Particular thanks to Beatrice Boesch and Janet Nurnberg for their efficient and cheerful administrative support.

Samir Ziada Thomas Staubli McMaster University Canada Switzerland

Hochschule Technik + Architektur Luzern

XIV


Organization

ORGANIZING COMMITTEE

M. Balda, Czech Republic E! W. Bearman, United Kingdom I? Billeter*, Switzerland J. Bruggeman, Netherlands J. Fitzpatrick, Ireland H. R.Graf*, Switzerland E Hara, Japan E.de Langre, France l? Monkewitz*, Switzerland A.Oengoeren*, Switzerland M. F! Paidoussis, Canada W. Polifke*, Switzerland K. Popp, Germany D. Rockwell, USA H. Schmid, Austria T. Staubli*, Switzerland - Co-chairman D. S.Weaver, Canada S. Ziada, Canada - Chairman

* Local Organizing Committee

xv

Flow induced Vibration, Ziada & Staubii (eds) 0 2000 Balkema, Rotterdam, ISBN 90 5809 129 5

Acknowledgements

The organizing committee is grateful to the following corporations, organizations and institutions for the generous financial support they provided to FIV2000:

ABB Fachhoc hsc hule Zentralsc hweiz Kanton Luzern maxon motor McMaster University PILATUS AIRCRAFT LTD Pilatus-Bahnen Schiffahrtsgesellschaft Vienvaldstattersee Schweizerischer Verein des Gas- und Wasserfaches (SVGW) Stadt Luzern SULZER INNOTEC AG Swiss Aircraft and Systems Enterprise Corp Swiss National Science Foundation V-ZUG AG

XVI

Vortex- induced vibration

This Page Intentionally Left Blank


A three-dimensional model for wave and vortex-induced vibration of deepwater riser pipes

Jos6 A. Ferrari Jr Petrobras SA - E&R Rio de Janeiro, Brazil

Peter W. Bearman Department of Aeronautics, Imperial College, London, UK

ABSTRACT: A model to predict the wave and vortex-induced response of offshore risers is proposed. It couples dynamic solutions for the two planes of vibration and takes account of the hydroelastic interactions which occur between the structure and the fluid when the riser oscillates under wave and sheared current flows. A conventional approach for modeling fluid/structure interaction along the riser is adopted and a finite- element structural model is used, coupled to suitable hydrodynamic models for in-line and transverse loading. The method addresses the multi-mode response problem and correctly predicts the attenuation experienced by modal waves as they travel along the riser. The motion of the platform is shown to have a large influence on the response amplitudes of risers.

1 INTRODUCTION

Vortex-induced vibration (VIV) is a potential problem affecting many types of offshore structure including: production and drilling risers, conductors, pipelines, moorings, tethers of tension leg platforms, spar platforms and the members of jacket structures. In addition to increased stresses due to VIV, a h r - ther concern is the increased likelihood of collisions between adjacent risers. The most common structural form used in offshore engineering has a circular cross-section and this is the shape treated throughout this paper.

Vortex generation and shedding is observed to occur for slender bluff structures in a current, in waves and in a combined flow of waves and a current. The flow around structures in oscillatory motion is usually described by the Keulegan Carpenter number (KC) and either th: Reynolds nu_mber(Re) or the beta parameter. KC = UTD, where U is the peak velocity during a cycle, T the time period of the_flow oscillation and D is the body diameter. Re = UD/v, where v is kinematic viscosity and beta = ReKC = D2/vT. At very low KC there is no flow separation and at KC values when separation is first observed the flow remains symmetric. Sarpkaya (1986) carried out experiments on circular cylinders at low KC numbers, for different p values, and reported separation first occurring for KC numbers of order l . Observations by various researchers (e.g. Bearman, et al. (1 98 1)) indicate that fluctuating transverse forces due to vortex generation only become signifi-

cant at KC values in excess of about 4. In contrast to pure oscillatory motion, the flow always separates in the presence of a steady current, above very low Re. For offshore risers, KC values range txpically between 10 and 100 and Re is around 10 , hence the flow will separate and vortices will always be a dominant feature of the flow.

The search for an analytical tool to predict VIV of risers reliably and within a practical time scale has occupied the offshore industry over at least the last 20 years. The use of CFD techniques is particularly difficult and challenging, not just because Re is high but also because length to diameter ratios of typical risers may be several thousand. The resulting response is multi-modal with travelling structural waves. 3D solutions are now available but direct simulation is limited to low Re and small aspect ratios. Many approximate techniques have been proposed to predict particular aspects of VIV response and numerous model tests have been carried out to support specific aspects of the methods. The main purpose of many of these tests has been to provide empirical coefficients for the analytical formula- tions. Strictly, the application of these methods is restricted to the cases where the model test conditions match those of the hll-scale.

Detailed descriptions of recent models, and a comparison of their results for specific base cases, have been presented by Larsen, et al. (1985). As a general conclusion it can be stated that modeling travelling waves, and the spatial attenuation which takes place along a riser, is a key requirement for a

successful formulation. Hence a model which adopts a standing-wave response philosophy is expected to provide less reliable results than one which uses a travelling wave assumption.

In this paper we present a prediction model which accounts for the attenuation experienced by modal waves when they travel along the riser. This method addresses the multi-mode non lock-in problem and employs existing hydrodynamic models in the description of the fluid/structure interaction. The ideas of Morison are used in the in-line direction and the quasi-steady model for transverse forces due to vortex shedding by Bearman et a1 (1984) is employed in the cross-flow direction. A Quasi 3-D model is proposed in the sense that the hydrodynamic loading along the riser varies according to the response and there is coupling of the hydrodynamic loading between the two planes of vibration.

2 STRUCTURAL MODEL FOR A RISER

The so-called rigid riser is a long pipe between the ocean floor and an offshore platform, which is not designed to withstand large in-line deflections. They possess high axial stiffness and relatively high flexural rigidity. This means that the Bernoulli-Euler as- sumptions of elementary beam theory can be applied to give for the in-line direction, for example:

where x is the in-line displacement experienced by the riser at a position z along its axis, EI(z) is the pipe flexural rigidity or bending stiffness, T(z) is the effective tension of the riser which includes the effects of internal and external pressure, N(z,t) is the normal force per unit length acting on the riser, and pA is the mass per unit length of the riser and its contents.

The Galerkin weighted residual method is applied to determine the solution to equation (1). The analysis is restricted to two dimensions for the sake of simplicity and the riser pipe is idealized as an as- sembly of beam elements. Each element involves six degrees of freedom, two translations and one rota- tion at each end. The offshore riser is considered free to rotate at both ends and an in-line displacement can be applied at the top node to represent platform offset. For firther details see Ferrari (1998).

3 IN-LINE HYDRODYNAMIC FORCE MODEL

The conventional way to predict loads on risers due to wave and current flows is to use Morisons Equa- tion. The accuracy of this approach depends largely

on the selection of suitable hydrodynamic coefficients. There are clear advantages in using Mori- sons Equation, but it cannot predict the oscillatory forces due to vortex shedding. This has proved to be a serious shortcoming, particularly for loading in the transverse direction, i.e., orthogonal to the wave propagation direction. In this work we use Morisons Equation for in-line loading.

Substituting the relative velocity formulation of Morisons equation into the general equation of motion in its matrix form we obtain:

where

U and U are the water particle velocity and acceleration induced by waves, p, is the density of sea water, CM is the inertia coefficient, CA = C, - 1 is the added mass coefficient,C,is tl .? drag coefficient, L is the riser element length, [K] is the global stiffness matrix, [M] is the structural mass matrix and [B] is the structural damping matrix. The water particle kinematics are calculated using Stokes 5th-order theory which provides the wave kinematics up to the free surface.

Equation (2) is solved by numerical integration in the time domain (Newmark J3 method with p =1/4) so that the fluid drag term can be evaluated accu- rately. The predictions for in-line displacements of the riser have been compared to those presented in an API Bulletin (1992) for several test cases. This Bulletin presents a comparison of the performance of a number of codes developed to predict both .he static and dynamic in-line response of rigid risers. Good agreement was obtained between the results in the API Bulletin (1992) and those provided by the solution of equation (2) (refer to Ferrari (1 998)).

4 VORTEX-INDUCED TRANSVERSE FORCES ON FIXED CYLINDERS

4.1 Fixed Cylinder in an Oscillatory Flow Bearman et a1 (1984) studied the unsteady transverse forces on fixed cylinders in oscillatory flows and presented a quasi-steady model to predict the transverse forces at high KC numbers (> 20). The main assumption in their model is that instantaneously the

4

wake flow is similar to that behind the same body in a steady flow. Predictions provided by this model were compared with experimental measurements of the transverse forces generated by an oscillatory flow (KC numbers ranging from 5 to 53). Generally speaking, the agreement between predicted and measured transverse forces is good at high KC but deteriorates for low values of KC, where the vortex wake in oscillatory flow differs significantly from that behind a cylinder in steady flow.

In oscillatory flows, since the velocity varies with time, the vortex shedding frequency f, also varies. Bearman et a1 propose :

(4)

and U is the instantaneous velocity of the oscillatory flow and St is the Strouhal number. T, is the average number of vortices shed per second from one side of the cylinder. The integral in (4) must be evaluated for each half cycle of the flow, i.e., for each positive and negative cycle of U. The following expression for the transverse force per unit length of the cylinder was derived by Bearman et al:

F,(t)=-pU2DC7cos(27lf,t+~) 1 2 ( 5 )

where is the amplitude of the transverse force coefficient, cp is the phase angle between the force and the oscillatory velocity. It should be noted cp and c a r e assumed constant over a half cycle of the flow. In addition, the model assumes that S, is constant. According to Bearman et al, this model fits experimental data extremely well at high KC numbers and reproduces the amplitude and frequency modulation seen in time histories of the transverse force.

4.2 Fixed Cylinder Subjected to an Oscillatory Flow Superimposed on a Sheared Current

I n this case the instantaneous velocity is assumed to be given by

u=u, cos(kx- ot)kU, (6) and this velocity is used in equation (5) to predict the unsteady transverse forces. The current velocity in equation (6) can be varied with depth to represent a current profile (shear current). However, it should be noted that at each depth, z, the value of the vortex shedding frequency corresponding to the steady flow, must be recovered at the end of each half cycle of the oscillatory flow.

Substituting equation (6) into equation (4), we obtain

and to refers to the beginning of a half cycle of the oscillatory flow. The following expressions shedding frequency of the flow are obtained:

)i f, sf U =--- I Kc (sin(h-wtr1 I- ~ D 271(t-t0) J for a positive cycle of the oscillatory jlow

[for anegative cycle of the oscillatory jlow

T T rr

and the modulus sign KC is defined as - 0 1 D

for the

is used

to ensure a positive value for the shedding frequency, whatever the instantaneous direction of the flow. Substituting the shedding frequency values given by eq. (8) and the flow instantaneous velocity (eq. (6)) into eq. ( 5 ) , and assuming appropriate values for the transverse force coefficient and phase, provides the transverse forces.

In a purely oscillatory flow there is no need to have a continuity of phase between one half cycle and the next. However, with a superimposed current the vortex shedding is continuous and there can be no jumps in the phase or magnitude of vortex shedding at the ends of flow oscillation cycles. Hence the current acts as a long term memory for the combined flow and this property can be used to determine the variation of phase during a flow oscillation cycle. To illustrate this, we have from eq. (5)

t var ies from 0 tot (9)

where the subscripts c and c+w refer to only current and wave-current flows respectively, p cor- responds to the transverse force phase associated with the uniform flow and cp represents the phase for the superimposed flow. Equating the two forces in eq. (9) at the beginning and at the end of each half cycle leads to:

At the beginning of the half cycle (t = 0 , U = U,)

5

pini=2iT-t*+p V J C D

At the end of the half cycle (t = % , U, = U,,,)

Here the time t* represents the instant when the oscillatory flow changes its direction. ARer defining the transverse force phase at the beginning and end of a half cycle, intermediate values for cp can be assessed assuming linear interpolation within the time interval %.

5 VORTEX-INDUCED FORCES ON RESPONDING RISERS

The quasi-steady model can also be used to estimate the transverse forces on responding risers. The riser may be vibrating both in-line and transverse to the flow. Hence, the relative velocity in the in-line direction has to include the riser in-line velocity x . The average of the instantaneous relative velocity is used to calculate the vortex shedding frequency at each half cycle of the oscillatirig flow. Note that this formulation for the transverse forces is expected to give better results at high KC numbers, where

l v r o +,IT and vfl is the amplitude of the KC = D

relative oscillating flow.

Similarly for the flow past a fixed cylinder, the average shedding frequency of a riser oscillating in the in-line direction is as follows:

The integral in equation (1 2) must be calculated in a step by step fashion for each half cycle of the wave flow. Then, the VIV force will be

F,,, = -p ( (U- i )+u , )* D ~,cos(2nf,t1+cp) 1 2 (13)

t varies from o to T* where the phase cp is calculated by the same procedure as described for the fixed cylinderhiser case. In the absence of a steady flow, the same phase cp is assumed at each halc cycle of the oscillating relative flow. The period T represents the half period corresponding to each half cycle of the relative oscillating flow. It should be noted that FVIV does not account for any reaction of the fluid when the riser responds.

6 QUASI 3-D DYNAMIC ANALYSIS OF OFFSHORE RISERS

6.1 Local Fluid-Structure Interaction

The relative velocity form of Morisons equation has been applied successhlly to the case of a riser moving in the in-line direction. For a riser also oscillating transverse to the flow, the relative velocity should include the effect of both components of the riser velocity. Then, we have the following formulation for the resulting force per unit length acting on the riser:

where

du F, = c A - + c ~ A ~ ~ v ~ ~ ( u - X ) - c,A,X (in-line a t

direction) (15)

Fy = FVIV - cDADlvr ly - C A A I Y (transverse fluid reaction

direction) (4 where FVIV is the vortex-induced vibration force as defined in equation (13) and I v ~ ~ = , / ( L I - x)* + y 2 . Note that U represents the summation of wave and current velocity.

6

Now, we can write the full transverse force per unit length acting on a responding riser:

L

Note that a Morison-type formulation has been adopted to represent the fluid reaction force oppos- ing the transverse motion of the riser. Rajabi et al. (1984) considered a similar approach for the resis- tive force in the transverse direction apart from the damping force, which they took as C D ~ D l j l Y . We believe that the present model for the transverse damping force is more realistic since it accounts for the influence of the in-line relative flow on the attenuation of the transverse response.

With respect to equation (17), in model tests hydrodynamic coefficients are not assessed independ- ently and reported values of the transverse force coefficient for responding cylinders represent the total transverse force, including viscous damping and inertia effects. Hence, the amplitude of the transverse force coefficient c, in equation (17) has been obtained from experiments on stationary cylinders where the fluid-structure interactions are not taken into consideration.

6.2 Coupling between In-Line and Transverse

The general equation of motion solved in the in-line direction can also be solved for the transverse direction. The basic matrices of the system, namely mass, stiffness and structural damping, can be assessed in the same way as has been done for the in-line direction. The added mass coefficient as well as the dominant modes and corresponding damping factors will have to be consistent with the transverse behaviour of the riser.

Concerning the method of solution, the same numerical integration scheme used to solve the in- line problem, namely Newmark p, is employed to solve the general equation of motion in the transverse direction. Now, the equations to be solved can be written as

Planes

[M], x + [B], X +[K], x =

where IV,I=d(u+U, -k)2 + y 2 and the subscripts x and y represent in-line and transverse directions respectively. The solution of equations (18) and (19) is it- erative around the riser velocity values x and y respectively. In addition, the relative flow in the in- line direction, U and au/a,, depend on the riser in-line displacement x; which is not known until equation (18) is solved. Hence the solution procedure entails three different loops of iteration as described in Fer- rari (1998).

7 NUMERICAL APPLICATIONS

A code, based on the foregoing methodology, has been developed to predict riser response. Before solving the dynamic problem itself, the fundamental matrices of the system (mass, structural damping and global stiffness) are assembled and the natural frequenciedmodes of vibration are evaluated in both directions. Equations (18) and (19) are then solved in the time domain and in a coupled fashion. Values for the riser displacements and velocities in both directions, bending moments, shear forces, stresses at the top node, and so on, are stored during the run so that peak values as well as time histories are available. As an example, the case of a production offshore riser subjected to only waves is analysed as fo 110 w s :

Riser pipe outside diameter : 0.25m Riser pipe inside diameter : 0.21 lm Modulus of elasticity 210,000,000.0 kN/m2 Density of sea water: 1025 kgf7m3 Density of the fluid in the riser bore: 800 kgf7m3 Density of the riser wall material: 7700 kgf/m3 Water depth: 300.0m Riser length: 320.0m Top Tension: 500 kN (1.5 times the riser self- weight) Static offset: 0.0 (vertical riser) Wave height: 6.3m Vessel motion amplitude/phase : 2.0m / 90.0 degrees

Base Case : Production Riser

Corresponding period: 8.5s

Before carrying out the fu l l calculation, a short run was performed in just the in-line direction to evaluate the eigenmodedfrequencies and the KC and Re numbers along the riser. Then, the following data related to modeling the in-line and transverse forces were selected :

du C A - + a t

X)- C,A,x

1 p ( t u - i ) + U , ) 2 D Ctcos(2nfst+cp) - C,A,y -

Drag and inertia coefficients in both directions (constant along the riser): 1 .O and 1.6 respectively. Dominant modes: 1st and 2nd in the in-line direction and 5th and 6th in the transverse direction with damping factor < = 0.02.

7

Transverse force phase for positive and negative cycles of the relative velocity: 20' and 200' (constant). Strouhal number: 0.2. Amplitude of the transverse force coefficient: 1.5.

3 4 5

The riser was modeled with 50 elements equally spaced below the Mean Water Level and 4 elements equally spaced above this level. The time step for the dynamic analysis was 0.05s so as to trace the frequency modulation of the transverse forces. A run time equivalent to 30 times the wave period was useded. These parameters were adopted following the sensitivity analysis described by Ferrari (1 998).

Table 1 shows the natural periods for the first 15 modes of vibration of the riser and the KC numbers required to excite each eigenmode. It can be noted that the 5th and the 6th modes were considered as the dominant modes of vibration of the riser in the transverse direction. A multi-modal, non-lock-in response is expected since the relative velocities, and the shedding frequencies, will have a wide variation along the riser span.

The amplitude of the transverse force coefficient is consistent with the results presented by Sarpkaya & Isaacson (1981) for fixed rough cylinders. An average KC number of 22 was adopted for the estimate of C, and an average Reynolds number of 1 .5~10' (beta G 7000) was used. According to Bearman (1988), experiments on circular cylinders in waves suggest that postcritical flow is reached at consid- erably lower Reynolds numbers than in steady flow. In addition, turbulence in the incoming flow and a rough surface due to marine growthlcorrosion shift the flow regime boundaries downwards. These findings support the decision to consider a postcritical value for the drag coefficient in the present example. The values of CD and CM in the present example were taken from Bearman (1988).

4.6 9.2 3.4 12.5 2.6 16.3

Table 1 - Natural Periods of the Riser

9 10 11

Note that the difference of phase Acp between positive and negative half cycles is kept constant and equal to 180'. Although phase cp is considered constant, the spanwise variation of the relative flow velocity causes frequency changes at the different stations. This means that the transverse forces are uncorrelated spanwise between elements. The values for the transverse force phases are consistent with the findings of Bearman et al. (1984). Figure 1 shows results for the maximum transverse displacements experienced by the riser. The results are presented for three different conditions: 1) ne- glecting the fluid damping (C,, = O.O), 2) considering a one-dimensional approach for the fluid damping (F,~ =CD,AD1ylj), and 3) considering a two- dimensional approach in which the in-line flow contribution to the damping force is accounted for in the transverse direction (FDt =CDLADIVrIj). The main

purpose of these plots is to assess the influence of the fluid damping on the transverse response. We believe that the transverse response calculated for the 2D case is the most reliable one. A computation time equivalent to 4 x T, was needed to obtain the steady-state solution in the damped cases whereas a period around 20 x T, was required for the undamped one. We can note that the fluid damping is re- sponsible for reducing the response to a third of its undamped magnitude on the upper part of the riser, and by around 30% on the lower part. Along the responding riser the KC number ranges from 15 to 30 and the dominant modes of vibration are the 5* to the gth.

As can be seen in Figure 2, the 2nd mode of vibration is the dominant one in the in-line direction. This result is consistent with the natural periods presented in Table 1 (T, = 8.5). It is also verified that the in-

1.3 32.7 1.1 38.6 1 .o 42.5

number

12 13 14 15

0.9 47.2 0.85 50.0 0.8 53.1 0.75 56.7

if:l 28.3

Figure 1 - Envelope of the Maximum Transverse Response

8

line response is not influenced by the transverse response of the riser. The same base case was run in the in-line direction only and the envelope of the maximum response was nearly the same as the one presented in Figure 2 (difference < 3%).

Figure 3 shows the time history of the transverse displacements at two nodes of the riser, namely node 3 7 and node 8 located 2 16m and 42m above the sea- bed respectively. These nodes were chosen because the riser experienced large responses around these stations. It can be seen from this figure that the riser responses are periodic as expected. Planar views of the riser responses at nodes 8 and 37 are shown in figure 4. The different forms of the responses in the two directions are clearly visible.

Figure 5 shows the maximum transverse and in- line responses for the same test case, apart from the fact that there is no translation at the top node. Fig- ure 5 has to be compared with Figures 1 and 2, This simulation is aimed at assessing the influence of the motion of the platform on the transverse forces and responses. It can be seen from Figure 5 that the re- duction of the in-line motion at deeper stations of the riser is not as significant as it was before. This is because the velocity of the riser is smaller in this case and hence the fluid damping is less. For fbrther discussion of these results see Ferrari and Bearman (1 999).

We have considered as a basic assumption that vortex shedding is not locked on to the riser frequency of vibration. It is assumed that the shedding frequency depends exclusively on the in-line relative flow velocity, Strouhal number and riser diameter. This is because multi-mode responses rather than single-mode ones are expected to take place along the riser. The basis for this assumption is that there is a wide variation of KC number along the riser span, i.e., many modes are likely to be excited. Un-

fortunately we do not know how to set precisely boundary that must exist between the single-mode lock-in case and the multi-mode non-lock-in one.

Figure 2 - Envelope of the Maximum In-Line Response Figure 5 - Envelope of the Maximum Response Considering No Top Node Translation

9

However, some researchers, and particularly Vandi- ver (1993), have already dealt with this matter in experimental studies for the case of a flexible cylinder subjected to a shear flow. According to Vandiver (1993), the shedding frequency will be locked to the cylinder frequency and a single-mode response is expected if: 1) the current shear fraction is small, i.e., AUJV,,a, < 0.3 where AUc is the difference between the maximum and the minimum velocity in the profile and U,,a, is the maximum velocity; 2) the potential number of responding modes within the shear excitation bandwidth is greater than 10. In the present numerical example, we have around 12 modes being excited and a shear fraction factor equal to 1.0. Therefore, the assumption that the riser will experience multi-mode response under non- lock-in conditions is consistent with previous work.

Ferrari (1998) presents results for the same base case considering a shear flow (current) superimposed on the wave flow. Once more multi-mode responses are expected due to the wide variation of KC number. With respect to the magnitude of the predicted transverse response, Guerandel et al. (1995) present experimental results for a single, flexible cylinder, representing a riser in 500m of water depth and subjected to a combined loading due to waves and a shear current. An underwater video tracking technique is used to measure directly the in-line and transverse responses experienced by the riser model. No translation at the model top node was considered in this study. Although not discussed by the authors, it is supposed that the riser model vi- brated under non-lock-in conditions. As a general conclusion, the results for the maximum transverse displacement envelopes are very similar to those found using the method presented here, reaching peak values as high as 1D. Hence, these results help to provide some validation of the prediction model presented here.

8 CONCLUSIONS

The main contribution of this work is the development of a prediction model for the response of deepwater risers which couples dynamic solutions for the two planes of vibration of the structure. The model accounts for the hydroelastic interactions which take place between the structure and the fluid when the riser oscillates in a wave flow and a combined wave and sheared current flow. The hydrodynamic forces in the in-line direction are predicted by Morisons equation and those in the transverse direction by a quasi steady model due to Beannan et al (1984). The results for an offshore riser, based on this methodology, for realistic wave and current flow conditions seem to be very promising. Most importantly, the type of multi-mode response observed in experimental studies is reproduced.

9 ACKNOWLEDGEMENTS

J A F expresses his gratitude to the Brazilian oil company, Petrobras, for sponsoring his maintenance and studies at Imperial College, London.

REFERENCES

API Bulletin 16J 1992. Bulletin on Comparison of Marine Drilling Riser Analyses, 1st ed., August 1.

Bearman, P.W. et al. 1981. The Role of Vortices in Oscilla- tory Flow about Bluff Cylinders Proc. Intern. Synzposium in Ocean Engineering, Vol. 2, pp. 621-35, Trondheim, Norway.

Beannan, P.W. et al. 1984. A Model Equation for the Trans- verse Forces on Cylinders in Oscillatory Flows, Applied Ocean Research, Vol. 6 , No. 3, pp. 166-172.

Beannan, P. W. 1988. Wave Loading Experiments on Circular Cylinders at Large Scale, Proc. 5th Int. ConJ on the Be- haviour of Offshore Structures, Trondheim, Norway, pp.

Ferrari, J.A. 1998. Hydrodynamic Loading and Response of Offshore Risers, Ph.D. Thesis, University of London, June.

Ferrari, J.A. & P.W.Bearman 1999. A Quasi 3-D Model for the Hydrodynamic Loading and Response of Offshore Ris- ers, Paper JSC-I 14, ISOPE 99, Brest, France.

Guerandel, V.L et al. 1995 Marine Riser Model Tests in Waves and Currents, Proc. of the Intern. Conference on Offshore Mechanics and Arctic Engineering, pp, 363-73, Copenhaguen, Denmark.

Larsen, C.M. & K.H.Halse 1995. Comparisons of Models for Vortex-Induced Vibrations of Slender Marine Structures, Proceedings of the 6th Conference on Flow-Induced Vibra- tion, A.A. Balkeina Publishers, pp. 467-82, London, UK.

Rajabi, F et al. 1984. Vortex Shedding Induced Dynamic Re- sponse of Marine Risers, Transactions of the ASME, Vol. 106, pp 214-221, June.

Sarpkaya, T. & M.Isaacson 1981. Mechanics of Wave Forces on Offshore Structures, 1st ed., Van Nostrand & Reinliold Company.

Sarpkaya, T. 1986. Force on a Circular Cylinder in Viscous Oscillatory Flow at Low Keulegan-Carpenter Numbers, J. FluidA4echanics, 165, pp. 61-71.

Vandiver, J.K., 1993. Dimensionless ParaIneters Important to the Prediction of Vortex-Induced Vibration of Long, Flexi- ble Cylinders in Ocean Currents, Journal of Fluids and Structures, Vol. 7, pp. 423-455.

471-487.

10

Flow induced Vibration, Ziada & Staubli (eds) 0 2000 Bakerna, Roeerdarn, lSBN 90 5809 129 5

Vortex induced vibrations measured in service in the Foinaven dynamic umbilical, and c o ~ p ~ s i o ~ with predictions

G.J. Lyons - Department of ~ e c h a n i c a ~ ~ngineering, U n ~ ~ e ~ s ~ ~ College L o ~ d ~ n , UK J. K.Vandiver - Department of Ocean Engineering, Massachusetts Institute of Technology, Cambridge, Mass., USA

C. M. Larsen -Department of Marine Structures, Norwegian University ofScience and Technology, Norway ~ . ~ A s ~ c o ~ b ~ - BP Amoco, S u n b u ~ UK

ABSTRACT: The Foinaven U~bilical ~oni tor ing System ( F U ~ S ~ has been provided to measure the stresses in a subsea umbilical of the Foinaven floating production facility. The FUMS provides high quality data for vortex induced vibrations. A novel approach has been adopted to clearly demonstrate the variation with time of the contributions of umbilica~ excitation at mooring system, wave. and vortex shedding frequencies. The Foinaven case is of particular interest owing to the existence of strong currents which may be substantially sheared, and the large number of modes of vibration which may be induced by vortex shedding. This paper presents some of the responses measured, and makes comparison with vortex induced vibration predictions. The predictions include drag coefficient amplification resulting from VIV which is important for the design of future L~~nb~licaIs owing to its influence on the total hydrodynamic load, and on station keeping of the vessel. Recommendations for future design practice are included.

1 INTRODUCTION

For floating production systems the umbilical is a critical component for the control of subsea opera- tions. It provides electrical and hydraulic power and signals to the subsea components in the system from the surface vessel. It will be subject to dynamic and static loads throughout its anticipated 20 years or so service life.

Figure 1 . Umbilical co~~gura t ion

Relatively high frequency Vortex Induced Vibrations are a possible source of damage to marine risers, urn- bilicals, and moorings. However, this phenomenon has not apparentIy been significant for flexible risers and umbilicals in the North Sea. The reasons for this have been considered to be in part owing to the complex shape of the umbilical (compared with a vertical riser), substantial structural and hydrodynamic damping, and less extreme currents.

The Foinaven ~mbilical ~ o n i t o r i ~ g System (FUMS) measures the stresses in one of the two subsea umbilicals of the relatively deep water (465m) Foinaven floating production facility (FPSO), Figure 1. This is a new area of exploitation extended from the North Sea to the Atlantic margin, west of Shetland.

High quality data recorded by FUMS have been available from March 1997 to date, The system provides information on the relative effects of vessel motion, and current (including VIV) on the stresses in the DC2 dynamic umbilical. The stresses are derived using a specialIy designed curvature sensor which acts at the entry of the ~ b i l ~ c a l into the un- derside of the vessels connection turret (Fig.2), and nieasureme~it of top tension. Additional sensors niotiitor turret heave, pitch arid roll (aligned with the umbilical), turret surge and sway, heading, and significant wave height.

Some of the effects observed on Foinaven have already been described in several technical papers (Lyons et al, 1998a,b,c). This paper presents the results from further studies funded by the Norwegian Deepwater Programme (NDP) comparing some of the data gathered and predictions using global, and VIV models. Of particular significance is the impact of VIV on Cd amplification for the extreme response of the umbilical in the global sense. Estimates of this are presented, as are recommendations on various aspects of design, including structural damping, aiid lift coefficient values.

2 CURVATURE MEASUREMENT

The curvature sensor is deployed in a spare 5/8" hydraulic control hose in the umbilical. It measures curvature at three locations. in two orthogonal directions (X aiid Y), in the region of the umbilical bend stiffener (see Fig 2), where L1 refers to the upper- most sensing position, L2 the middle, and L3 the lower. It is terminated at the FPSO connector deck

measurement region below the vessel keel. Curvature with direction is detected using strain gauges config- ured in pairs for each location to ensure tension and temperature effects are eliminated.

2.1 Data acquisition

This utilizes two separate 486 processing units. The main unit samples 16 channels of analogue data at 20 samples/sec/cliaIlliel, for a total of 8192 samples per channel. Hence each sampling period is just under 7 minutes. The 20 Hz sampling was chosen to deal with the anticipated maximum 3Hz vortex induced vibration frequencies. The 7 minute sampling period being adequate to cope with several cycles of surge and sway.

Subsequent to acquisition, the data are processed. which includes marking for quality, calculation of statistical values, incrementation of rainflow tables. and storage of time series if preset thresholds have been exceeded. This takes of the order of three minutes to complete, before acquisition of the next set of data. Hence statistical data are updated at approximately 10 minute intervals.

level and extends through the vessel depth to the 2.2 Unzbilicnl Conjigzrvution

A significant feature of this design is that 4 outer layers of armouring were required to achieve a compatible weightldiameter ratio with the associated ten risers for which an analogous behaviour might be expected, Fig 3. This figure also shows

Figure 3. Umbilical cross-section

Table 1 -- Umbilical physical details

Figure 2.Umbilical exit from turret

12

the hydraulic and chemical injection hoses, and central electrical conductors. Buoyancy (flotation) modules of 0.92m diameter distributed as shown in Figure 1 enable the pliant wave configuration. Physical details are given in Table 1 .

2.3 Measured VIV

Statistical data gathered includes the maximum, mean, minimum, and standard deviations of meas- urands mentioned above (Fig 4). Also Acoustic Doppler Current Profiling (ADCP) data are available at 10 minute intervals, and so it is possible to make comparison with these and the curvature measurements which had been recorded in frequency bands representing mooring system, wave (vessel motion), and VIV frequencies (Table 2). VIV for the Foinaven umbilical configuration occurs principally owing to current effects. This is evi- denced from the M3 and M4 standard deviation re-

Table 2 Frequency Filtering Bands

sponses for curvature which are closely correlated with measured currents. Large motions of the FPSO have been shown in this study as capable of inducing VIV. However, there is some evidence from the data that vessel motions tend to reduce the current induced VIV response, which was anticipated since the relative flow is more disrupted (less correlated) along the umbilical with time, Lyons & Fang (1991), and Fang & Lyons (1991).

Note that frequency band VIV2 (M4) should be viewed as the extra component of M3, and so where M3 peaks appear to be capped it is appropriate to look to M4 for the missing cap (Fig 5 ) .

3 VIV ANALYSIS

3.1 General

There are several VIV computer programs available which have the potential for prediction for flexible catenary risers and umbilicals. Each has its own at- tractions, and limitations. Of those available from the authors of this paper, Fang and Lyons (1991), and Vandiver (1999), it was decided to use SHEAR 7 as part of the NDP study since an earlier version of this code had been used in the design of the umbilical system, and was of benefit in providing subsequent comparison with the measured data.

Figure 4. Hs, heave, curvatures M1, 2, 3 variations with time, 9-Jun-97 to 29-Jun-97

13

Figure 5 . Curvature L1 sde components, and current speed

3.1.1 Transfer functions jroni curvature to dis- placenzent

Since curvature is measured by the FUMS rather than displacement, it was necessary to convert these to displacements for comparison with the VIV predictions. This required that finite element models be generated for the bend stiffener, and the umbilical. The RIFLEX (for the umbilical) and RISANA (for the bend stiffener) analyses, Fylling et al (1 998), and Larsen (1 997), provided transfer functions from curvature to VIV amplitude response.

Figure 6. Curvature frequency transfer function

variations with time, 20-May-97 to 9-Jun-97.

One such frequency dependent transfer function which relates to the L1 sensing location for the VIV frequency range considered may be expressed as:

Amplitude(m) = 1 9.0 * curvature(m- ' )/frequent y (Hz) This is shown graphically in Figure 6.

The transfer functions were used to estimate the response spectra of the VIV amplitude using the VN curvature spectra obtained from the time series data. An example displacement spectrum is shown in Fig- ure 7. In general the upcrossing frequency of the response amplitude in all VIV cases analyzed fell between 0.45 and 0.63 Hz. At the typical average upcrossing frequency of 0.475 Hz seen in the VN data, the transfer function value at L1 is approximately 40m2.

RMS values of the VIV response were computed by taking the square root of the sum of the mean square values in the x and y directions at each sensor location. The mean square values were obtained by integrating the displacement response spectra.

3.1.2 Data analysis

Time series data were available at only selected times. Statistical summaries of the data were available at all times and were used to find interesting

14

cases for time series analysis and to understand problems with various seiisors.

The time series records of curvature have static and dynamic content from a variety of causes, including waves, current, and vessel motion. The VTV contribution to the dynamic response may be sepa- rated from that due to other causes by high pass filtering. The curvature response spectra were computed after high pass filtering the curvature time series data with a 9 pole Butterwortli filter with a cutoff frequency of 0.25Hz. All contributions to the measured curvature above this frequency were assumed to have come from VIV origins.

Dla Y97A15.5 U Dlb Y97B156 P>Q

D4 Y98A314 B D7 Y97.4161 A,B,C,D

D8a Y97A256

3.2 Case Studies for comparison

June 4 1997 1438 June 5 1997 0925, 0935 Nov. 10 1998 0028 June 10 1997 1410, 21,3 1,4 1 Sept. 13 1997

3.2.1 Summary ojall cases

The cases which were analyzed were chosen so as to cover a variety of field conditions, including calm (Dla), current only (Dlb and D7), heavy seas only (D4), and combined seas and current (D8a, b). The D4, heavy seas only case, was chosen to verify that significant vessel heave could alone cause VIV in the absence of significant current. This was verified.

The availability of measured current profiles from ADCP current meters provided the basis for prediction of VIV response. These predictions included both A/D,,, a id Cd amplification. SHEAR7 was not designed to handle unsteady, motion-induced VIV. Therefore only steady currents were assumed in the analysis. From estimates of the A/D,,,,, response, the drag coefficient amplification factor was computed and passed back to the global FEM analysis for further consideration, but is not reported herein.

The cases were selected to give a variety of sea- state and current combinations. As shown in Table 3, the AID,,,, response did not exceed 0.2 diameters of the umbilical for any of the cases. This result is based on curvature measured at the L1 sensor. If one used an average of the measurements from the L1 and L2 sensors, then the maximum A/Dr,,,, would

Table 3. Summary of cases examined for VIV

Filename and time T

E974256 I

RMS fHeave(m) /Hs(m)

rj = 0.04

Hs = 0.8 H = 0.04

Hs = 1.9

H = 1.6

Hs = 2.1 H = 0.16

Hs = 2.4

H = 0.64

Hs = 7.2 H = 0.72

Hs = 7.1

Max. C u rr-e n t (mis)

10.1

0.66

0.305

0.86

0.56

0.84

VIV AID,,,, upcross ing Cd,ainp No VIV

0.17 0.57 Hz 1.52

0.16 0.45 Hz 1.50 0.18 0.47 Hz I .54

0.15 0.47 Hz 1.48 0.14 0.63 Hz 1.46

have been around 0.27 diameters. However, a review of the perforniance of the three sensing positions showed that at L1 to be more likely to be the most accurate. The cases with both current and significant vessel heave, have slightly smaller response than with current only. For example compare D l b to D8a or D7 to D8b. From this it is postulated that a rapidly time varying velocity component tends to disrupt the VIV that would normally result from a steady current. All cases with vessel motion, including those with current, have less total VIV response than the current only cases. This conclusion is based on a small sample of data, and further analysis will be carried out in future to further verify this assertion.

Current profiles for cases Dla, Dlb, D7 and D8 are shown in Figures 8, 9, and 10. At the Foinaven

Figure 7. Example spectrum Figure 8. Current profiles cases Dla and D1 b

15

Figure 10. Current profiles cases DSa, DSb, and D7

site the currents are relatively fast with significant shear owing to the effects of the Gulf and Arctic streams at different levels. The twice-daily effects of changing current are evident from the VIV response and current velocities, as seen in Fig 5 where the M 3 and M4 standard deviation (dynamic components) of curvature VIV peaks and troughs match with those of the 10-minute averaged mean water depth current speed. Whilst the current does swing through 360 degrees the predominant effect is from SW to NE and vice versa. This is in the direction of the plane in which the umbilical lies. For this reason the VIV analyses presented here only consider the modal response appropriate to excitation in this plane.

3.3 Results

3.3.1 A/D Predictions

Figure 11 is a typical predicted A/D,,,, response for the case D7. In this analysis VIV excitation was assumed to exist only on the small diameter portion of the umbilical without flotation. This is the top 62% of the length of the umbilical. In its present form,

relative position

Figure 1 1 . Prediction of A/D,,,,,

SHEAR7 cannot model two different hydrodynamic diameters in the same run. Therefore, in the analysis shown here, the entire umbilical was assumed to have the diameter 0.185m. This was assumed, because the strongest VIV excitation will occur in the strongest current region, near the top of the umbilical where the diameter is small. This will result in high frequency vibration waves travelling down the riser from the top as shown in Figure 12. This figure is an example of the travelling wave behaviour of riser response to excitation near the top end. In this example a vortex shedding frequency of 0.55 Hz was used, corresponding to a current speed of approximately 0.55 d s , such as in Case Dlb. Since the SHEAR7 model was not intended to model excitation on the flotation region this prediction is quantitatively valid only in the bare portion of the umbilical. However, once the vibration waves enter the flotation region they will damp out quickly as shown in the figure. Also since the important component of current velocity is that normal to the umbilical, the effective velocity re- duces with increasing umbilical inclination, which is substantial for much of the flotation region. This re- duces the possibility of V N excitation for the frequencies identified. Thus in the region with flotation, Figure 12 is qualitatively accurate. The very irregular diameter in the flotation region will suppress significant VIV. Although some response from the flotation region is possible, lack of experimental data for such structures prevents mak- ing a quantitative prediction of the amount. Some facts are known about the likely V N characteristics of the flotation region, which are described here. First, any VIV would be at very low frequency, due to the large diameter of the floats and due to the low expected Strouhal number. The Strouhal number would be low due to the very low aspect ratio of the floats. The region will also have significantly greater hydrodynamic damping. Due to the low expected VIV frequencies from the flotation region, it would not be possible to separate curvature caused by waves and vessel motion from that resulting from

16

Figure 12. Typical prediction of wave propagation in response to VIV excitation near the top end at 0.55Hz

VIV on the flotation. In examining the curvature data for cases with current and calm seas no obvious VIV was present at frequencies that would be associated with VIV on the flotation. It is concluded here that for this particular umbilical, VIV on the flotation region is not a significant factor in VIV related fatigue or drag coefficient enhancement.

In this particular analysis the structural damping was set at 0.5% of critical. An added mass coefficient of 2.0 (with respect to the small umbilical diameter) was used to give the overall umbilical the correct average mass per unit length, due to the presence of the flotation. This was necessary because the mass of the flotation was not included in the structural mass per unit length in the input data. An added mass coefficient of 1 .O was also tried with no significant difference in the predicted VIV behaviour.

SHEAR7 requires use of a lif t coefficient table to determine the lift coefficient. The lift coefficient is a function of the response A/D, which makes iteration necessary. This lift coefficient table was increased by a factor of 1.08 to raise the predicted response to 0.18 diameters. Thus the lift coefficient for the dominant responding mode was approximately 0.65. It was necessary to use a rather low structural damping, 0.5% and an enhanced lift coefficient table (increased by 8%) to predict the level of vibration estimated for this umbilical, from the curvature records.

Other runs were conducted, but with very little difference in results, and are not included herein.

4 DRAG ENHANCEMENT

4.1 Drag Coeflcient Amplijication Factor

rms AID Figure 13 Cd amplification

This is an empirical curve, derived from measured drag forces on a flexible steel pipe (L=22.86 m, D = 0.04m) with VIV in modes 1 to 3, Vandiver (1983). There are many other similar curves published by a variety of authors, though most used a single degree of freedom oscillator or driven cylinder data from laboratory experiments. The above curve was for a flexible cylinder excited by tidal current. The figure gives an amplification of 1.5 to 1.55 for AID,,,,, from 0.15 to 0.2 diameters. Figure 14 shows the Cd amplification factor for the A/D,,,, shown in Figure 1 1. From this a conservative amplification factor of 1.6 appears to be appropriate for the small diameter regions. The mean Cd on the floats as con- figured on Foinaven will only increase by a small amount due to VIV related motion. An amplification of 1.2 in the flotation region should suffice for design purposes. The staggered layout of the flotation on the Foinaven umbilical is very good for discouraging VIV. The uneven diameters, and the low aspect ratio (1engtWdiameter) of the floats, both serve to reduce the VIV. Little data exists to support a more precise prediction of Cd in the flotation region.

1.7 I I I I ) . I , , ,

The amplification factor calculated is a function of A/Dr,,s as given below and plotted in Figure B 13.

Figure 14. Estimate of the mean Cd amplification factor (maximum current case D7)

CD,nnlp =1.0 + 1.637(A/ DRMs)0.65

17

5 DISCUSSION

5.1 Modeling

The geometry of pliant wave umbilicals (and risers) is more complex than that of vertical tensioned members. Consequently these umbilicals require more attention to the way in which they are modeled if accurate responses (particularly VIV) are to be obtained.

Particular attention must be paid to: distribution of mass and hydrodynamic diameter identification of potential VIV power-in active regions identification of potential VIV energy dissipative regions

*

Q

The principal concern for these is with the provision of buoyancy aids.

The limitations of the computer code used must be recognized. Potential difficulties of such codes include (but are by no means limited to):

ability to cope with variations in properties along the umbilical (eg mass, diameter, relative flow velocity, various coefficients, effective tension, bending stiffness) ability to generate or import eigenfrequen- cies/modes ability to deal with a sufficiently high number of modes ability to consider added mass as an anisotropic mass, both for the bare umbilical and for sections with buoyancy modules (This modelling feature is of importance when computing eigenfrequen- cies/modes of the umbilical in its plane, relevant for current perpendicular to this plane).

Having recognized the deficiencies it may still be possible to proceed to provide meaningfd results. One strategy which has been successfully used, has been to consider the regions with and without buoyancy aids differently. In this arrangement the bare umbilical alone is assumed to be capable of VIV excitation. The section with flotation collars is assumed to be capable only of energy dissipation.

Whilst it may not be possible to model the umbilical with variations in diameter it may be possible to perform the analysis as follows:

The entire umbilical may be assumed to have the bare diameter. This may be assumed, because the strongest VIV excitation will occur in the strongest current region, near the top of the umbilical where the diameter is small. This will result in high frequency vibration waves travelling down the riser from the top. If the VIV model is not intended to model excitation on the flotation region this prediction is quantitatively valid only in the bare portion

of the umbilical. However, once the vibration waves enter the flotation region they will damp out quickly.

The very irregular diameter in the flotation region will suppress significant VIV. Although some response from the flotation region is possible, lack of experimental data for such structures prevents a quantitative prediction of the amount.

Some facts are known about the likely VIV characteristics of the flotation region:

Any VIV would be at very low frequency, due to the large diameter of the floats and due to the low expected Strouhal number.

* The Strouhal number would be low due to the very low aspect ratio of the floats.

0 The region will also have significantly greater hydrodynamic damping.

Owing to the low expected VIV frequencies from the flotation region, it would not be possible to separate curvature caused by waves and (if suspended from a vessel) its motion, from that resulting from VIV on the flotation section since these would be of similar frequency.

In examining the curvature data from Foinaven for the response from currents with calm seas, no obvious flotation region VIV was present. For this particular umbilical, VIV of the flotation region is not a significant factor in VIV related fatigue or drag coefficient enhancement.

5.2 Structural Damping

Estimates of likely suitable structural damping values have relied upon tests on flexible risers, and cables, which have been of relatively short lengths (tens of metres).

One of the authors (Vandiver) has conducted tests on a 13m long horizontal power cable sample, with a diameter of 0.14m and under a tension of 40kN. The resulting damping ratio was approximately 1.5% at 3 Hz for vibration of the first asym- metric mode.

Lyons and Fang reported on a wide range of tests on a 10m long, 2 inch flexible riser in free-hanging and catenary configurations from 1'' to 3'd mode of vibration at ambient and elevated temperatures, Fang & Lyons (1992). This indicated damping ratios of 2.5 to 25%. These values are too high for application for umbilicals which are substantially longer than this, and at higher tensions. It has been shown, Fang & Lyons (1996), and Brown et a1 (1996), that damping decreases at higher tensions, and depends on wavelength and curvature (mode). This work has shown the rate at which damping increases with length appears to be very rapid. However, it is difficult to extend the existing data to accurate values of structural damping for lengths much in excess of 50m.

18

Previously Lyons & Fang (1991) showed that fluid damping over passive regions is expected in general to be an order of magnitude higher than structural damping. As a consequence the choice of a low value of structural damping is appropriate and should lead to reasonable estimates of VIV.

It is recommended that until more data become available that a value of structural damping of 0.5% of critical should be used based upon the VIV analyses used to match prediction with measured data from the Foinaven umbilical.

5.3 Lift CoefJicients

Experience with one VIV code (SHEAR7) which requires use of a lift coefficient versus A/D table to determine the lift coefficient showed that it was necessary to increase the lift coefficient table by a factor of 1.08 to match predicted with measured (Foinavenj values. This resulted in a peak lift coefficient of 0.65. It is recommended that other codes which utilize lift coefficients should consider this enhancement also.

6 CONCLUSIONS

prime driver for VIV. VIV as a consequence of large wave and vessel motion has also been demonstrated for the Foinaven umbilical. This at present is considered to occur in practice less often for deep water umbilicals than current driven VIV, and so its effect is likely to be of less overall significance to design.

There is some evidence from the Foinaven data that vessel motions tend to reduce the current induced VIV response, which was anticipated since the relative flow is more disrupted (less correlated) along the umbilical, Fang & Lyons (1991 j.

Many of the comments are directly transferable to flexible risers. From these it is evident that the FUMS has enabled us to be in a better position in respect of how we should be dealing with both global and VIV analyses.

The work reported here has been limited in the extent of the number of cases examined. Plainly there is a lot to be learnt from the data such as FUMS can provide. Technology is advancing, and it is now possible to design a data gathering system which provides more detail on the umbilical response through more of its length. It is hoped that in the not-too- distant future we will have been able to implement this technology with pliant wave umbilicals and risers to improve our understanding of their response in greater detail.

6.1 VIV response

7 ACKNOWLEDGEMENTS VIV response of umbilicals is significantly different to that of tensioned vertical risers. Whereas for tensioned vertical risers the VIV response is found in the frequencies from the fundamental upwards, for dynamic umbilicals (and flexible risers) the VIV modes of interest are much higher, and are unlikely to include those modes near the fundamental. Gener- ally the response of umbilicals will be more compli- cated than that of tensioned vertical risers, in particular owing to their global geometry. It is possible that some of the modes of VIV interest will be in the frequency band of vessel motions and waves. It is more likely that those VIV responses of interest will however be higher than these.

VIV response for the Foinaven dynamic umbilical has been shown to be timewise broadband with no ND,,, values greater than 0.2 (based on the most reliable L1 sensing location) having been measured to date. However, most VIV energy has been shown to be within an identifiable range (0.45 to 0.63 Hzj which is outside the range normally considered for wave and vessel motion. VIV response has been shown to generally decrease beyond this identifiable range with increasing mode number, Lyons et al (1 998).

VIV can regularly exist if the current conditions permit. This is true in deeper waters where the dis-. ruptive effects of waves and vessel motion are less. It may generally be expected that current will be the

This work has been carried out as part of the Nor- wegian Deepwater Programme Dynamic Umbilical Guidelines activity, administered by BP Amoco. The authors wish to thank the sponsors of the for per- mission to publish the work reported herein.

8 REFERENCES

Brown, D. T., G. J . Lyons & H. M. Lin. 1996. Effects of Catenary Mooring Line Damping, Final report for ULOS Managed Programme of research in Reducing the Uncer- tainties in Offshore Structures, EPRSC/MTD.

Fang, J. & G. J. Lyons. 199 1 . Application of a General Predic- tion Method for Vortex Induced Vibrations of Catenary Risers, ISOPE, Proc Ist International Offshore and Polar Engineering Conference, Edinburgh, ISBN 0-9626 104-8-8 (Vol III) , pp 354-361.

Fang, J . & G. J. Lyons. 1992. Structural Damping Behaviour of Flexible Risers. J. Marine Structures, Design, Construc- tion and Safety, Vol 5 , pp 162-192.

Fang, J . & G. J. Lyons. 1996. Structural damping of tensioned pipes with reference to cables, J. Sound and Vibration, Vol 193(4), pp 891-907.

Fylling, I.J., C.M. Larsen, N. S ~ d a h l , E., Passano, A. Bech, A.G. Engseth, H. Lie, & H. Ormberg. 1998. RIFLEX - User's Manual. Revised version 1998-09-24. MARINTEK report STF70 F952I8, Trondheim.

19

Larsen, C. M. 1997. RISANA - a computer program for static and dynamic riser analysis, Input description, Department of Marine Structures, Norwegian University of Science and Technology.

Lyons, G. J. & J. Fang. 1991. Vortex Induced Vibrations cf Tensioned and Catenary Marine Risers. Paper C4 16/042, Proc, Intl Conf on Flow Induced Vibrations, Brighton, ISBN 0-85298-764-1, IMechE, pp 75-86.

Lyons, G. J., D. T. Brown, H. H. Cook, B. Walls, G. Bamay. 1998. The Foinaven Umbilical Performance Monitoring System - A New Approach. OTC 8883, Proc Offshore Technology Conference, Houston.

Lyons, G. J., H. H. Cook, & M. Ashworth. 1998. High fie- quency motion components in a large dynamic duty umbilical. Int Cony Hydroelasticity in Marine Technology, Kashiwagi et al Eds, RIAM, Kyushu Univ, Japan, 1998, ISBN 4-87780-001-8, pp 273-282.

Lyons, G. J., B. Walls, & H. H. Cook. 1999. Dynamic Um- bilical Performance - Results from Foinaven. Proc 3rd European Conference oti Flexible Pipes, Utnbiliculs and Murine Cubles - MARINFLEX 99, Bentham Press, Lon- don, ISBN 1-874612-29-3, pp 14.1 to 14.19.

Vandiver, J..K.. 1983. Drag Coefficients of Long Flexible Cyl- inders. Proc. Offshore Technology Conf, Paper 4490, Houston.

Vandiver, J.K. et al. 1999. User Guide for SHEAR7. Massa- chusetts Institute of Technology.

20


Predicting lock-in on drilling risers in sheared flows

J. Kim Vandiver Department of Ocean Engineering, Massachusetts Institute of Technology, Cambridge, Mass., USA

ABSTRACT: The kurtosis statistic is introduced as a sensitive tool for efficient preliminary analysis of flow- induced vibration response data. The kurtosis of sequential blocks of time series data allows one to distin- guish single mode lock-in events. The problem of predicting lock-in is then discussed. Instructional examples are introduced to illustrate the relative importance of flow-speed, power-in length and diameter, when attempting to predict single frequency dominance. When VIV from two different flow speed regions com- pete, it is shown that the ratio of the flow speeds cubed is an important indicator, as is the ratio of the square of the length of the power-in regions. Two example cases from measured response on a drilling riser in the North Sea are presented. SHEAR7 predictions are compared to measured results.

1 INTRODUCTION

I n the past two years BP/Amoco and the Norwegian Deepwater Project have made available vortex- induced vibration (VIV) data from four drilling risers in the North Sea. Although the velocity profiles have shown great variation in speed and direction with depth, single mode dominated response has been frequently observed. The observed single-mode response occasionally has lock-in properties similar to those seen under uniform flow conditions. In particular the onset of lock-in is accompanied by an increase in response amplitude, a dominance of a single frequency and development of constant response amplitude sinusoidal behavior. Finding and classifying these single frequency events can be a tedious exercise, requiring sorting through thousands of multi-channel records of response data. In this paper it is shown that the statistic known as kurtosis is especially useful in detecting lock-in events. By using this statistic it is possible to efficiently sort VIV response according to behavior.

Once lock-in and non-lock-in events are identified, the next step is to understand and explain the flow conditions and the structural dynamic properties that govern the occurrence of lock-in. The final step is to develop response prediction models that are able to predict response behavior.

This paper first presents the use of the statistic kurtosis in the identification of lock-in events. Sec-

ond, simplified instructional examples are used to reveal the relative importance of parameters, such as damping and length of the lock-in region. Third, the MIT VIV response prediction program SHEAR7 is used to predict vibration of a North Sea drilling riser, using measured current profiles. Comparisons are made between measured and predicted VIV.

The author thanks BP-Amoco for the Schiehal- lion riser data, and the Norwegian Deepwater Pro- gramme (BP-Amoco, Esso, Norsk Hydro, Saga, Shell, Statoil, Conoco, Mobil) for the Helland Han- sen Riser Data."

2 USING KURTOSIS FOR LOCK-IN IDENTIFICATION

Typical drilling riser response measurements are from instruments strapped to the riser at several different positions. Assume that x ( t ) is a zero mean, measured time series of transverse acceleration response, resulting from VIV. The kurtosis of x ( t ) is given by:

kurtosis =o, where ( ) = time average ( 1) Kurtosis is normally used to quantify deviation from Gaussian behavior. Zero mean Gaussian processes have a kurtosis of 3.0. Multiple frequency VIV response is often Gaussian in behavior, and tends to

(x')'

21

have kurtosis values of approximately 3. The occa- sional occurrence of a single frequency, lock-in event is characterized by steady sinusoidal behavior. When x(t)=sin(wt) , the kurtosis takes on the value of 1.5.

By plotting the kurtosis of a typical response measurement over time it is possible to quickly identify transitions from multi-frequency behavior to single-frequency, constant amplitude lock-in events.

Figure 1 is an example of VIV response data, taken every 48 minutes on the Scheihallion drilling riser in the North Sea. Time, spanning several days, is plotted horizontally in Figure 1. The maximum tidal current over all water depth is plotted with the kurtosis of the transverse acceleration response measured on the drilling riser at a position ZL = 1/8. z is the axial coordinate on the riser as measured up from the bottom, and L is the total riser length. This particular riser was in a water depth of 368 meters, and depending on the strength of the current re- sponded with VIV in the 1'' to 4'" modes. The first four modes all have substantial modal amplitude at W8.

Twice per day tides dominated the current and for several days in a row as shown in Figure 1, current conditions permitted one mode to dominate the vibration. At these times the response in the mode fa- vored by the current profile built up to the point that significant VIV at different frequencies from less favorable regions of the riser was suppressed. Wake synchronization was established in the power-in region and lock-in occurred. As a result, the response

grew in amplitude, was dominated by a single frequency, and the kurtosis approached 1.5, the ideal sinusoidal value. This is seen several times in Fig- ure I . Further discussion of the Scheihallion data is to be found in Cornut & Vandiver(2000).

This particular data covers a span of eleven days. The lowest line in the figure is the maximum current, which occurred in the entire water column, as measured by an acoustic Doppler profiler (ADCP). There is not a simple correlation between maximum current speed and lock-in events.

The prediction of a lock-in event requires exami- nation of each current profile to determine the power-in region of each possible contending mode. One must then have a means of comparing the relative strengths of each mode. When one mode has a dominant position, then lock-in occurs and other modes are squeezed out. The next section presents a model for the prediction of single mode dominance, based on structural dynamic properties and the flow profile.

3 PREDICTING MODAL DOMINANCE

3.1 Introdu

Flow Induced Vibration

Documents

Transcript of Flow Induced Vibration