Appendix A: Correction of Non-Dimensional Frequency for Actual Strouhal Number

The tests by Gopalkrishnan (1993) were carried out with a small cylinder. According to the standard curve for Strouhal number, as a function of Reynolds number, the Strouhal number, \(\mathrm {St}_G\), must have been approximately \(\mathrm {0.2}\) for these tests. If we have another Strouhal number and want to apply Gopalkrishnan’s results, we have to take this into account when calculating the non-dimensional frequency parameter.

General case: \(\mathrm {St}^*\)

Vortex shedding frequency

\[f_v^*=\frac{\mathrm {St}^*U^*}{D^*}\]

Non-dimensional frequency

\[\hat{f}^*=\frac{f_\mathrm {osc}^*D^*}{U^*}\]

Figure 1. Current, diameter and oscillating frequency for the general case in VIVANA.

Gopalkrishnan’s case: \(\mathrm {St}_G\)

Vortex shedding frequency

\[f_v=\frac{\mathrm {St}_GU}{D}\]

Non-dimensional frequency

\[\hat{f}=\frac{f_\mathrm {osc}D}{U}\]

Figure 2. Current, diameter and oscillating frequency for Gopalkrishnan’s data

The general case is assumed to be equivalent to Gopalkrishnan’s case if

\[\frac{f_\mathrm {osc}^*}{f_v^*}=\frac{f_\mathrm {osc}}{f_v}\]

The non-dimensional frequency can now be defined from parameters in the general case:

\[\frac{f_\mathrm {osc}^*}{\displaystyle \frac{\mathrm {St}^*U^*}{D^*}}=\frac{\displaystyle \hat{f}\frac{U}{D}}{\displaystyle \mathrm {St}_G\frac{U}{D}}=\frac{\hat{f}}{\mathrm {St}_G}\]

Hence

\[\hat{f}=\frac{f_\mathrm {osc}^*D^*}{U^*}\frac{\mathrm {St}_G}{\mathrm {St}^*}\]

By using this definition for the non-dimensional frequency in all operations we can use Gopalkrishnan’s data directly.

If we calculate the reduced velocity from the standard definition

\[U_R=\frac{U}{f_0D}\]

for a system with Strouhal number \(=\mathrm {St}^*\) that is different from \(\mathrm {St}_G\), and want to compare such values to values obtained from Equation (8). we have to correct \(\mathrm {U_R}\) in order to take care of the difference in Strouhal number. If we assume that the ratio \(\mathrm {\displaystyle \frac{f_0}{f_v}}\) should be constant, the correction will be as follows

\[U_R=\frac{1}{\hat{f}}\sqrt{\frac{\overline{m}+1}{\displaystyle \overline{m}+\frac{C_a}{C_{a0}}}}\]

With standard Strouhal number:

\[\frac{f_0}{f_v}=\frac{\displaystyle \frac{U}{U_RD}}{\displaystyle \mathrm {St}_G\frac{U}{D}}=\frac{1}{U_R\mathrm {St}_G}\]

At a different Strouhal number:

\[\frac{f_0}{f_v}=\frac{1}{U_R\mathrm {St}^*}\]

We can now define a modified \(\mathrm {U_R}\) that will give the same ratio. This must be as follows:

\[U_R^*=U_R\frac{\mathrm {St}_G}{\mathrm {St}}\]

By using this definition of the reduced velocity we can compare \(\mathrm {U_R^*}\) directly to values found from the standard relation between \(\mathrm {U_R}\) and the non-dimensional frequency in Gopalkrishnan’s experiments.