Line with floats: Axial drag coefficient

The line type wizard sets up the axial drag coefficient $\C{Da}$ for a line with floats using a similar approach to that for the normal drag coefficient.

Considering the derived line, with the additional buoyancy smeared along its outer surface, the drag force per unit length, when flow is axial to the line, is due solely to skin friction and can be expressed as \begin{equation} \label{fDa1} f_\mathrm{Da} = \frac12 \rho\, v^2 \C{Da} \pi d_{\mathrm{a}l} \end{equation} in which the reference drag area per unit length is the circumference of the base line (calculated from the axial drag diameter $d_{\mathrm{a}l}$), and where $\rho$ is the density of seawater and $v$ the magnitude of the flow velocity.

As for flow normal to the line, we can also express the drag force per unit length experienced by the derived line as the sum of the drag forces experienced by the floats and those experienced by the exposed part of the line not hidden by the floats \begin{equation} \label{fDa2} f_\mathrm{Da} = f_{\mathrm{Da}f} + f_{\mathrm{Da}el} \end{equation} In this case, however, as well as the drag due to skin friction there is an additional drag force from the exposed annulus on the end of each float. The reference drag area of this annulus is \begin{equation} \frac{\pi}{4} \frac{\left(d_f^2 - O\!D_l^2\right)}{s_f} \end{equation} leading to the total axial drag experienced by the floats \begin{equation} \label{fDaf} f_{\mathrm{Da}f} = \frac12 \rho\, v^2 \left( C_{\mathrm{Da}f1} \frac{\pi}{4} \frac{\left(d_f^2 - O\!D_l^2\right)}{s_f} + C_{\mathrm{Da}f2} \pi \frac{d_fl_f}{s_f} \right) \end{equation} The axial drag experienced by the exposed part of the line is \begin{equation} \label{fDael} f_{\mathrm{Da}el} = \frac12 \rho\, v^2 C_{\mathrm{Da}l} \pi \frac{d_{\mathrm{a}l} (s_f-l_f)}{s_f} \end{equation} Substituting from expressions (\ref{fDaf}) and (\ref{fDael}) into (\ref{fDa2}), and equating (\ref{fDa1}) and (\ref{fDa2}), we arrive at the final formula for the axial drag coefficient of the derived line type \begin{equation} \C{Da} = \frac{1}{d_{\mathrm{a}l} s_f} \left\{ C_{\mathrm{Da}f1} \frac14 \left(d_f^2 - O\!D_l^2\right) + C_{\mathrm{Da}f2} d_f l_f + C_{\mathrm{Da}l} d_{\mathrm{a}l} (s_f-l_f) \right\} \end{equation}