Line with floats: Added mass coefficients

The line type wizard sets up normal and axial added mass coefficients for a line with floats in much the same way as normal drag coefficients.

Normal added mass coefficient

Applying a similar analysis to that for drag coefficients to the added mass per unit length in the $x$-direction of the derived line \begin{equation} A\!M_\mathrm{x} = \rho\, \frac{\pi}{4} O\!D^2 \C{ax} \end{equation} in which the reference volume is the volume of the derived line type and $\rho$ the density of seawater, we arrive at the corresponding formulae for added mass coefficients for the derived line type.

For a base line type of the homogeneous pipe category\begin{equation} \begin{aligned} \C{ax} &= C_{\mathrm{an}f} \frac{d_f^2}{O\!D^2} \frac{l_f}{s_f} + C_{\mathrm{an}l} \frac{O\!D_l^2}{O\!D^2} \frac{s_f-l_f}{s_f} \\ \C{ay} &=\ \sim \end{aligned} \end{equation} For a general category base line type \begin{equation} \begin{aligned} \C{ax} &= C_{\mathrm{an}f} \frac{d_f^2}{O\!D^2} \frac{l_f}{s_f} + C_{\mathrm{ax}l} \frac{O\!D_l^2}{O\!D^2} \frac{s_f-l_f}{s_f} \\ \C{ay} &=\begin{cases} \, \sim & \text{if $C_{\mathrm{ay}l}=\, \sim$} \\ C_{\mathrm{an}f} \cfrac{d_f^2}{O\!D^2} \cfrac{l_f}{s_f} + C_{\mathrm{ay}l} \cfrac{O\!D_l^2}{O\!D^2} \cfrac{s_f-l_f}{s_f} & \text{otherwise} \end{cases} \end{aligned} \end{equation}

Axial added mass coefficient

The added mass coefficients follow in a similar way to $\C{ax}$ above. The reference volumes for the derived line type and for the floats and exposed part of the underlying base line are the same in axial flow as in normal flow, so we can simply take either of the above expressions for the added mass coefficient $\C{ax}$ in normal flow and replace the coefficients for normal flow with those for axial flow \begin{equation} \C{aa} = C_{\mathrm{aa}f} \frac{d_f^2}{O\!D^2} \frac{l_f}{s_f} + C_{\mathrm{aa}l} \frac{O\!D_l^2}{O\!D^2} \frac{s_f-l_f}{s_f} \end{equation}