Excitation Force Model

The excitation force in CF and IL direction at a given position on the structure is defined as the component of the hydrodynamic force that is in phase with the response velocity at the same position and for the CF and IL direction respectively. The force on an element with length \(\mathrm {\Delta L}\) is given by

The frequency of the force in CF direction follows the vortex shedding frequency, while the IL force acts at the double frequency.

The excitation force coefficient \(\mathrm {C_{e,CF/IL}}\) is a function of the non-dimensional amplitude \(\mathrm {A_{CF/IL}/D_H}\) and the corrected non-dimensional frequency \(\mathrm {\hat{f}_{c,CF/IL}}\). Note that for the same response condition \(\mathrm {\hat{f}_{c,IL}}\) will be identical to \(\mathrm {\hat{f}_{c,CF}}\) even if the IL response takes place at twice the CF frequency.

It is well known that VIV is a self-limiting process, which is easy to understand by realizing that \(\mathrm {C_e}\) becomes negative if the response amplitude exceeds a certain value. \(\mathrm {F_e}\) will hence change from being an excitation force to become a damping force.

A complete excitation force model consists of

The bandwidth is referred to as "the excitation range", and the part of the structure where the non-dimensional frequency falls within this range is referred to as "the excitation zone", see also Excitation zone. Figure 1 illustrates this concept for a vertical riser that has a buoyancy zone with increased hydrodynamic diameter. The excitation zones for the lowest response frequency are indicated. Note that damping may still take place within the excitation zone if the response amplitude becomes so large that the excitation force coefficient becomes negative.

Excitation force coefficients, excitation range and damping coefficients are defined as default values in the program, but these are valid for a smooth circular cylinder. The user is therefore allowed to specify her own data, which in particular is important to account for helical strakes or cross sections with satellite pipes as found on drilling risers. The following rules apply:

Note that the program allows use of different concepts and sets of curves for each segment along the structure.

The default excitation coefficient model for cross-flow VIV in VIVANA is based on the coefficients found by Gopalkrishnan (1993). However, some modifications to these curves were made in order to include the experience from flexible beam tests. Gopalkrishnan’s map of the excitation coefficient in a non-dimensional amplitude and frequency plane is shown on Plot of contour curves for CF excitation coefficient in an amplitude / frequency map, from Gopalkrishnan (1993).

Instead of defining the excitation force coefficients as a two-parameter function of amplitude and frequency, VIVANA’s built-in model applies a set of parameters that defines the coefficient as a function of the amplitude. The parameters are in turn given as functions of the frequency. This is done in order to make it easy to modify the model on the basis of new information without being able to define the complete set of data as shown on Plot of contour curves for CF excitation coefficient in an amplitude / frequency map, from Gopalkrishnan (1993).

Figure 2 shows how the CF excitation coefficient is defined as a function of the non-dimensional response amplitude for a given non-dimensional response frequency. The curve is assumed to have a maximum value (horizontal tangent) at \(\mathrm {B}\), meaning that \(\mathrm {AB}\) and \(\mathrm {BC}\) can be given as two second order polynomials when the three points \(\mathrm {A}\), \(\mathrm {B}\) and \(\mathrm {C}\) are defined.

The numerical values for \(\mathrm {A}\), \(\mathrm {B}\) and \(\mathrm {C}\) coordinates as functions of the non-dimensional frequency are defined in Figure 3 for cross-flow VIV. Appendix B gives the same values in a table.

Note that the excitation force coefficient will define damping for larger response amplitudes than \(\mathrm {A/D_{C_e,CF=0}}\) within the excitation zone. The curves give a reasonably good approximation to Gopalkrishnan’s data also for negative coefficients.

Examples of coefficients used in the program are shown on Figure 4. Each curve on the figure represents the CF excitation coefficient as function of the response amplitude for a specific value of the non-dimensional response frequency.

Gopalkrishnan (1993) has published results from forced harmonic in-line motions in uniform flow. An excerpt of these results are presented in terms of contour plots for the IL excitation coefficient, \(\mathrm {C_{e,IL}}\), as functions of non-dimensional frequency and amplitude ratio, see Figure 5. The \(\mathrm {C_{e,IL}=0}\) contour line defines the ideal IL lock-in condition simply because zero excitation force means that there is no energy transmission between the cylinder and the fluid - which also is the case for an undamped cylinder under steady state lock-in.

The excitation range for IL response is defined as \(\mathrm {0.2<\hat{f}<0.9}\) and is indicated on Figure 5. The full range of available data from Aronsen’s experiments is included, even if damping is seen to take place for all amplitude values at low and high \(\mathrm {\hat{f}}\) values. The reason for this decision is that direct use of Aronsen’s data will give a better damping model for in-line vibrations than use of the Gopalkrishnan model - which is based on forced CF motion tests. Damping at lower \(\mathrm {\hat{f}}\) values than \(\mathrm {0.2}\) is not likely to take place since the response must be expected to shift to a higher eigenfrequency. At the other extreme, \(\mathrm {\hat{f}>0.9}\), we have response at very low flow velocity, which can be modelled as still water damping. Hence, IL and CF vibrations will have the same damping for such cases.

Figure 6 shows the parametric IL excitation coefficient curve. This is slightly different from the CF case since the coefficient is assumed to be zero for zero amplitude. Parameters that define the position of the points \(\mathrm {A}\), \(\mathrm {B}\) and \(\mathrm {C}\) on Figure 6 are given as functions of the non-dimensional frequency on Figure 7. The curves that are defined from these parameters will be close to what would be found directly from the contour curves on Figure 5. Note that these curves are valid for pure IL response and must not be used for cases with simultaneous response in both directions. Numerical values for all points on Figure 7 are found in Appendix B.

IL response is known to be significantly larger in combination with CF than for the pure IL case. Hence, the coefficients that were found from pure IL experiments (Section 3) can not be used for combined IL/CF cases. Gopalkrishnan (1993) proposed to describe the IL amplitude as function of dominating mode order. However, the review of existing data presented by Gopalkrishnan (1993) could not verify this assumption.

As a preliminary solution for describing the IL excitation coefficient for combined response, we have decided to use the results from the PhD work of Gopalkrishnan (1993). The amount of data is far less than what is needed to cover all cases of practical interest, and no true interaction between IL and CF is accounted for. The excitation coefficient curve has the same shape as for CF (Figure 2), and the values of the 3 points that define the curve for a given value of non-dimensional frequency is given in Figure 72. Numerical values for these points are found in Appendix B.

Note that these data must be considered as preliminary. The curves might be changed as soon as new data becomes available.

A general excitation coefficient model in VIVANA allows for easy input of model test results or results from CFD calculations. Such data can be applied by use of two different formats:

An example of the second alternative is illustrated on Figure 9. These curves are valid for a specific design of helical strakes, and are seen to give damping (negative excitation coefficients) for all amplitudes and frequencies. The coefficient for a specific combination of frequency and amplitude is found by interpolation.

Use of curves as shown on Figure 9 can adequately handle cases with up to approximately \(\mathrm {75~\%}\) coverage of VIV suppression devices. For these cases the bare pipe controls the VIV behaviour (frequency, mode etc.), and the section with strakes will provide a significant damping instead of excitation on a bare riser.

If strakes are attached to a larger part of the pipe, this section will take control of the VIV behaviour. Model tests for such cases indicate a different physical behaviour than for cases controlled by the bare pipe. This behaviour seems to be strongly dependent on pitch and height of the strakes. The responding frequencies and modes are generally lower than expected. VIV analyses by VIVANA of such cases are therefore not recommended.