Initial Calculations

The initial information concerning hydrodynamic parameters that is needed for application of the present theory is as follows:

  • The hydrodynamic diameter \(\mathrm {D_H}\), which is used in all calculations of hydrodynamic coefficients. For a circular cross section the hydrodynamic diameter is identical to the physical diameter, while data for other cross sections (like pipes with helical strakes or fairings) can be referred to a diameter that in principle is arbitrary.

  • The drag coefficient, which is used in the initial static analysis of the structure. This input value should be multiplied by the drag amplification factor that is calculated by VIVANA in order to find correct drag forces under VIV conditions. Note that combined use of RIFLEX and VIVANA makes it possible to calculate a new static condition based on amplified drag coefficients caused by VIV.

  • Strouhal number for all cross sections valid for a stationary cylinder, which may be a constant value or given as a function of Reynolds number. Each cross section type may have its own Strouhal number value or curve. The Strouhal number is used in an initial evaluation and to use built-in information correctly for the actual case. Note that the Strouhal number is always related to the vortex shedding frequency for a fixed cross section, and will not be directly used for calculation of the dynamic response frequency. The built-in curve for \(\mathrm {St(Re)}\) is recommended unless the user has better information for the actual cross section.

  • Added mass, given for all cross sections and valid in still water. Any variation along the structure is allowed since the finite element method is applied for solving the eigenvalue problem. The input value for added mass is used to calculate the eigenfrequencies of the structure in still water. The input values are normally not applied in the succeeding VIV analysis unless this option is explicitly specified by the user.

The initial analysis will consist of calculating some key parameters based on input values. All parameters will be linked to an element in the finite element model. The position of the midpoint of element number \(\mathrm {i}\) is assumed to be defined by a coordinate \(\mathrm {s}\) along the structure, and also in global coordinates. The user must provide the following parameters:

Environment℃al data:

  • \(\mathrm {U(z)\quad }\) Current profile \(\mathrm {[m/s]}\)

  • \(\mathrm {T\quad }\) Water temperature (global) [\(\mathrm {^\circ }\)]

For non-vertical structures an effective current velocity profile will be defined by the component of the velocity perpendicular to the structure, \(\mathrm {U_N}\). Note that RIFLEX may allow space varying current profiles, which may be used to include the effect from the boundary layer above an uneven seabed. This option can be used for VIV analysis of free spanning pipelines.

The water temperature is needed since the kinematic viscosity of water - and hence also the Reynolds number - depends on the water temperature. The Reynolds number will be used to calculate the local Strouhal number from built-in or user specified data.

Riser data:

  • \(\mathrm {D_V(s)\quad }\) Volume correct diameter, applied to define buoyancy \(\mathrm {[m]}\)

  • \(\mathrm {D_H(s)\quad }\) Hydrodynamic diameter, applied to define all hydrodynamic parameters \(\mathrm {[m]}\)

  • \(\mathrm {m(s)\quad }\) Mass per unit length of pipe including content \(\mathrm {[kg/m]}\)

  • \(\mathrm {C_a(s)\quad }\) Added mass coefficient for still water \(\mathrm {[-]}\)

  • \(\mathrm {C_D(s)\quad }\) Drag coefficient for non-vibrating structure \(\mathrm {[-]}\)

Note that all the parameters above must be constant within a segment, cfr. RIFLEX User Manual.

The Reynolds number \(\mathrm {Re}(s)\) can be calculated from the specified data according to this equation. The Strouhal number \(\mathrm {St}(s)\) must be known in order to find the vortex shedding frequency for a non-vibrating structure. \(\mathrm {St}\) can be defined in three different ways (confer also VIVANA User Manual):

  1. The built-in curve for the Strouhal number as a function of \(\mathrm {Re}\) can be used, see Strouhal number as function of Reynolds number in VIVANA.

  2. The user may specify an alternative curve for the Strouhal number as a function of \(\mathrm {Re}\).

  3. The user may specify a fixed value for \(\mathrm {St}\) (independent of \(\mathrm {Re}\)) valid for each riser segment

Note that some segments of a structure may apply alternative 3 with different fixed values, while others may apply different curves.

Having access to the mass and stiffness matrices, VIVANA calculates the eigenfrequencies and associated eigenvectors (mode shapes) valid for still water. Results are presented in terms of:

  • \(\mathrm {f_{0,CF,i}\quad i=1,2,...,N_{CF}\quad }\) Eigenfrequencies, CF \(\mathrm {[Hz]}\)

  • \(\mathrm {\phi _i\quad i=1,2,...,N_{CF}\quad }\) Eigenvectors, CF \(\mathrm {[-]}\)

  • \(\mathrm {f_{0,IL,j}\quad j=1,2,... ,N_{IL}\quad }\) Eigenfrequencies, IL \(\mathrm {[Hz]}\)

  • \(\mathrm {\phi _j\quad j=1,2, ... ,N_{IL}\quad }\) Eigenvectors, IL \(\mathrm {[-]}\)

[tm_initial_parameters] and Figure 1 show how the above mentioned parameters are presented by VIVANA. The map of vortex shedding frequency along the structure together with eigenfrequencies gives an indication of active frequencies and modes. However, note that these eigenfrequencies will be modified since added mass under VIV conditions is different from the still water added mass.

Initial parameters; Strouhal number, hydrodynamic diameter and normal current velocity along the structure.

tm vortex shedding map
Figure 1. Map of vortex shedding frequency and eigenfrequencies.