Introduction

1. Abstract

VIVANA is a semi-empirical program for prediction of vortex-induced vibrations (VIV) for slender marine structures subjected to ocean current. The analysis method is based on a finite element model using beam or bar elements. The static condition is found from a general non-linear formulation that allows for very large displacements, but moderate strains. The dynamic analysis follows the frequency response method, and is hence linear, meaning that the VIV response must have small amplitudes relative to the length of the structure. The solution is found by an iteration that ensures consistency between response amplitudes and excitation coefficients, and also between load and response phase.

The program is linked to the program system RIFLEX that offers modelling features for analysis of structures like anchor lines, tensioned risers, conductors, tethers, umbilicals, flexible risers, free spanning pipelines and pipelines during installation. VIV analysis can hence be performed for the same structural types. A general 3D formulation of the structural geometry is offered, and current may have an arbitrary variation in speed and direction along the structure.

Hydrodynamic coefficients for pure IL, pure CF and combined IL and CF response are implemented in the program. The CF coefficients are identical for the two relevant options, while pure IL analysis will be based on coefficients that are different from those used in the combined IL and CF analysis.

Fatigue damage is calculated by assuming simultaneously acting frequencies or time sharing between all possibly active frequencies. Drag magnification for CF response is calculated from empirical equations, while experimental data are directly applied for adjusting drag in the pure IL case.

tm title figure

2. Notation

All parameters are defined at first appearance. Throughout the manual CF denotes cross-flow and IL denotes in-line. The following definitions should in particular be noted:

  • \(\mathrm {D_H\quad }\) Hydrodynamic diameter of cross section \(\mathrm {[m]}\)

  • \(\mathrm {U_N\quad }\) Flow velocity normal to cylinder axis \(\mathrm {[m/s]}\)

  • \(\mathrm {f_v\quad }\) Vortex shedding frequency, fixed cylinder \(\mathrm {[Hz]}\)

  • $f_\{}$ Response frequency \(\mathrm {[Hz]}\)

  • \(\mathrm {f_0\quad }\) Eigenfrequency, still water \(\mathrm {[Hz]}\)

  • \(\mathrm {m\quad }\) Mass per unit length, dry cylinder \(\mathrm {[kg/m]}\)

  • \(\mathrm {m_{H,CF}\quad }\) Hydrodynamic mass per unit length, CF \(\mathrm {[kg/m]}\)

  • \(\mathrm {m_{H,IL}\quad }\) Hydrodynamic mass per unit length, IL \(\mathrm {[kg/m]}\)

  • \(\mathrm {\rho \quad }\) Density of fluid \(\mathrm {[kg/m^3]}\)

  • \(\mathrm {\nu\quad }\) Kinematic viscosity \(\mathrm {[m^2/s]}\)

  • \(\mathrm {\omega \quad }\) Frequency, index as for \(\mathrm {f}\), \(\mathrm {[rad/s]}\)

  • \(\mathrm {A\quad }\) Response amplitude \(\mathrm {[m]}\) .

Non-dimensional parameters

  • \(\mathrm {Re}\) Reynolds number, \(\mathrm {Re}=\frac{U_ND_H}{v}\)

  • \(\mathrm {St}\) Strouhal number, \(\mathrm {St}=\frac{f_vD_H}{U_N}\)

  • \(\mathrm {\hat{f}\quad }\) Non-dimensional frequency, \(\hat{f}=\frac{f_{\mathrm {osc}}D_H}{U_N}\)

  • \(\mathrm {U_R\quad }\) Reduced velocity, \(\mathrm {\displaystyle U_R=\frac{U_N}{f_0D_H}}\) .

Hydrodynamic coefficients and forces

  • \(\mathrm {C_L\quad }\) Average (static) lift force coefficient

  • \(\mathrm {C_{e,CF}\quad }\) Excitation force coefficient, CF direction

  • \(\mathrm {C_{a,CF}\quad }\) Added mass coefficient, CF direction

  • \(\mathrm {F_{e,CF}\quad }\) Excitation force, CF direction. Force in phase with local CF response velocity \(\mathrm {[N/m]}\)

  • \(\mathrm {F_L\quad }\) Static lift force, normally not considered for VIV

  • \(\mathrm {m_{a,CF}\quad }\) Hydrodynamic mass per unit length, CF \(\mathrm {[kg/m]}\) .

  • \(\mathrm {C_D\quad }\) Average (static) drag force coefficient

  • \(\mathrm {C_{e,IL}\quad }\) Excitation force coefficient, IL direction

  • \(\mathrm {C_{a,IL}\quad }\) Added mass coefficient, IL direction

  • \(\mathrm {F_{e,IL}\quad }\) Excitation force, IL direction. Force in phase with local IL response velocity \(\mathrm {[N/m]}\)

  • \(\mathrm {F_D\quad }\) Static drag force, Morison equation \(\mathrm {[N/m]}\)

  • \(\mathrm {m_{a,IL}\quad }\) Hydrodynamic mass per unit length, IL \(\mathrm {[kg/m]}\) .

Hydrodynamic force and mass

\(\begin{array}{llllll}\displaystyle F_{e,CF}(t)=\frac{1}{2}\rho \,C_{e,CF}D_HU_N^2\sin(\omega _{CF}t)&\mathrm {[N/m]}&&&\displaystyle m_{a,CF}=\rho \frac{\pi D_H^2}{4}C_{a,CF}&\mathrm {[kg/m]}\\\\\displaystyle F_{e,IL}(t)=\frac{1}{2}\rho \,C_{e,IL}D_HU_N^2\sin(\omega _{IL}t)&\mathrm {[N/m]}&&&\displaystyle m_{a,IL}=\rho \frac{\pi D_H^2}{4}C_{a,IL}&\mathrm {[kg/m]}\\\\\displaystyle F_D=\frac{1}{2}\rho \,C_DD_HU_N^2&\mathrm {[N/m]}\\\\\displaystyle F_L=\frac{1}{2}\rho \,C_LD_HU_N^2&\mathrm {[N/m]}\\\\\end{array}\)

3. Program structure

The purpose of the computer program VIVANA is to calculate the response of a slender structure excited by vortex shedding due to ocean current. This response type is often referred to as vortex-induced vibrations (VIV). VIVANA is linked to RIFLEX (see RIFLEX Theory Manual) as shown on Figure 1. RIFLEX is developed by MARINTEK and tailor-made for static and dynamic analysis of slender marine structures. The two RIFLEX modules INPMOD and STAMOD are always a part of a VIVANA program system, while other RIFLEX modules are not needed. RIFLEX can handle a large variety of slender marine structures such as tensioned and flexible risers, anchor lines, umbilicals, tendons, pipeline during installation and free spanning pipelines. Such structures may hence also be analysed by VIVANA. Examples of analyses and a brief description of the first version of VIVANA is given by Larsen et al. (2001).

tm vivana structure
Figure 1. The overall structure of VIVANA and RIFLEX.

A complete VIV analysis consists of:

  • An initial RIFLEX analysis using the INPMOD and STAMOD modules.

  • The VIVEIG module computes normal modes and eigenfrequencies.

  • Some initial key parameters are calculated in INIVIV.

  • The VIVRES module carries out the dynamic response analysis
according to the method described herein. Cross-flow (CF) and/or in-line
(IL) response can be calculated.

  • The VIVFAT module calculates fatigue damage based on the results
from VIVRES.

  • Finally, VIV magnified drag coefficients are calculated in VIVDRG.

4. Analysis options

The original VIVANA program was limited to analyse 2D structures in unidirectional current profiles. The current was restricted to act in the plane of the structure or perpendicular to this plane. This limitation has been somewhat relaxed. Version 4.0 and higher allow analysis of configurations with more arbitrary geometry. The long-term goal is to also handle 3D current conditions.

Three analysis options are offered; confer VIVANA User Manual, Input to VIVIANA, Control information. 1. Cross-flow (CF) response only

Structures that satisfy the original VIVANA limitations will have the
same response when using this option as the response predicted by the
original VIVANA program, if the same values for all parameters are
applied. The present analysis model is, however, fully 3-dimensional.

  1. Pure in-line (IL) response

    This option was introduced in VIVANA version 3.1, and applies hydrodynamic coefficients found from forced IL motion tests, see Larsen et al. (2001). This response type takes place for current velocities lower than the on-set of CF vibrations, which means that the reduced velocity related to the fundamental eigenfrequency is lower than 2.5. This response type is normally not of interest to marine risers, but can contribute significantly to fatigue of conductors and free spanning pipelines. The analysis procedure is the same for IL as for CF response.

  2. Combined CF and IL response

    This option was introduced in VIVANA 4.0. The response is calculated in two steps. Step 1 is the CF response analysis and applies the same method and hydrodynamic coefficients as option 1. The IL response is calculated in Step 2, and is assumed to take place at two times the CF frequency found in Step 1. The analysis method is the same as for pure IL (option 2), but the hydrodynamic coefficients are different. In general we will see larger IL response amplitudes for the combined IL and CF response than for pure IL.

A brief outline of the analysis procedure will be given in Method Overview, while the method for dynamic analysis is described in The Global Geometry Definition, The Frequency Response Method and Calculation of Response Frequencies. Details on added mass, excitation and damping coefficients are given in Multi Frequency Response, Fatigue Analysis describes all options for fatigue analysis. Two alternative options are applied, one based on simultaneously acting response frequencies, and another based on a time sharing concept. Finally, data and equations for calculation of drag magnification are given in Drag Coefficient Modification.

5. Additional analysis options

An updated static analysis may be carried out by a second use of STAMOD where magnified drag forces are introduced. This type of analysis will require a special input file for RIFLEX STAMOD. The file is produced by VIVANA and described in the RIFLEX User Manual. Use of this option on marine risers is demonstrated by Larsen et al. (2001).

It is also possible to carry out an improved dynamic analysis by using intermediate results from VIVANA in a non-linear time domain model. Hydrodynamic coefficients for excitation, damping and added mass are found for the actual case through an iteration process in VIVANA, and these parameters are used as input parameters in the time domain analysis. The purpose of this analysis option is to account for local non-linearities like interaction between a free spanning pipeline and the seafloor at the span shoulders. This option is not available in the standard VIVANA program since the time domain option requires use of the RIFLEX DYNMOD module. For further details, see Larsen et al. (2001).