Dimensionless Parameters

1. Reynolds number

The Reynolds number classifies dynamically similar flows, i.e. flows that have geometrically similar streamlines around bodies of identical shapes when the incoming flow direction is the same. The condition for similarity is that the ratio of inertia force to friction force is constant at all corresponding points. The Reynolds number at a position \(\mathrm {s}\) along the structure is defined as

\[ \mathrm {Re}(s,T)=\frac{U_N(s)D_H(s)}{v(T)}\]

where \(\mathrm {v(T)}\) is the temperature dependant kinematic viscosity, found from Faltinsen (1990), see Figure 1. The \(\mathrm {s}\) coordinate follows the length of the structure in its deformed position.

tm kin visc vs temp
Figure 1. Kinematic viscosity as function of temperature.

2. Strouhal number

The Strouhal number is related to the vortex shedding frequency \(\mathrm {f_v}\) for a fixed cylinder, and defined by

\[ \mathrm {St}=\frac{f_vD_H}{U_N}\]

Note that the vortex shedding frequency in the general case will change when VIV occurs, but the Strouhal number should not be referred to a vibrating cylinder. The Strouhal number is used for an initial evaluation and identification of a preliminary list of possible response frequencies.

tm strouhal vs reynolds
Figure 2. Strouhal number as function of Reynolds number in VIVANA.

Figure 2 shows the built-in curve for \(\mathrm {St(Re)}\) in VIVANA. The curve is valid for a circular cylinder with some roughness and is taken from Faltinsen (1990). The user may specify another curve or keep the Strouhal number independent of the Reynolds number.

The selected curve for \(\mathrm {St(Re)}\) (see Figure 2) is assumed to be the best possible alternative for the present use of the Strouhal number, namely to find an initial value for the response frequency in an iteration. Experience shows that even if the vortex shedding frequency for a fixed, smooth cylinder might be significantly higher in the critical flow regime than what is indicated on the curve, the response frequency will drop to a level more like the rough cylinder case.

The vortex shedding frequency along a non-vibrating structure can be found from

\[f_v(s)=\frac{U(s)}{D_H(s)}\cdot \mathrm {St}(s)\]

Note that the Strouhal number is used to correct the non-dimensional frequency in order to apply the built-in or user given curves for hydrodynamic coefficients for vibrating cylinders correctly.

3. Non-dimensional frequency

The non-dimensional frequency \(\mathrm {\hat{f}}\) is used as a controlling parameter for added mass and excitation force coefficients. The non-dimensional frequency is defined by

\[\hat{f}=\frac{f_{\mathrm {osc}}D_H}{U_N}\]

The built-in data for added mass and excitation force are given as function of the non-dimensional frequency. These data are found from experiments at a given Reynolds number, and hence also for a given Strouhal number. The data will, however, be applied for other flow conditions, which means that they must be corrected. This correction is automatically performed in VIVANA by correcting the non-dimensional frequency according to change of Strouhal number:

\[\hat{f}_c=\frac{f_{\mathrm {osc}}D_H}{U_N}\frac{\mathrm {St_E}}{\mathrm {St}}\]

\(\mathrm {St_E}\) is the Strouhal number valid for the experiments from which the applied coefficients are found, and \(\mathrm {St}\) is the actual Strouhal number (see Section 2). This correction will ensure that the ratio between the oscillation frequency and vortex shedding frequency for the fixed cylinder is the same for the actual application as for the empirical basis. If the built-in data are used, \(\mathrm {St_E}\) is defined as \(\mathrm {0.2}\), and the correction will be performed accordingly. See also Appendix A.

Note that:

  1. Both built-in and user specified parameters for added mass and excitation coefficients are assumed to be valid for \(\mathrm {St_E}=0.2\) and corrected accordingly.

  2. The non-dimensional frequency for CF and IL components is referred to the respective oscillation frequencies. Hence, \(\mathrm {\hat{f}_{IL}=2\hat{f}_{CF}}\) for the same flow condition since the IL response frequency always is assumed to be two times the CF frequency.