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Line with floats: Normal drag coefficient |
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Line with floats: Normal drag coefficient |
The line type wizard sets up the normal drag coefficients $\C{Dx}, \C{Dy}$ for a line with floats. The precise formulae for these depends on the category of the base line type, though the derivation is much the same regardless of category.
The drag force per unit length of the derived line type when flow is normal to the line's axis in the local $x$ direction can be written \begin{equation} \label{fDx1} f_\mathrm{Dx} = \frac12 \rho\, v^2 \C{Dx} d_{\mathrm{n}l} \end{equation} in which the reference drag area per unit length, normal to the flow, is given by the normal drag diameter of the base line type, $d_{\mathrm{n}l}$, and where $\rho$ is the density of seawater and $v$ the magnitude of the flow velocity.
But 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 part of the line which is not hidden by the floats \begin{equation} \label{fDx2} f_\mathrm{Dx} = f_{\mathrm{Dx}f} + f_{\mathrm{Dx}el} \end{equation} where $f_{\mathrm{Dx}f}$ denotes the drag forces experienced by the floats and $f_{\mathrm{Dx}el}$ those experienced by the exposed parts of the line.
Taking the first of these, the contribution to the total drag force from the floats is \begin{equation} \label{fDxf} f_{\mathrm{Dx}f} = \frac12 \rho\, v^2 C_{\mathrm{Dn}f} \frac{d_fl_f}{s_f} \end{equation} where the term $d_fl_f/s_f$ is the normal drag area of the floats.
Now, the form of the second term $f_{\mathrm{Dx}el}$ depends on the base line type category.
A base line type of the homogeneous pipe category has isotropic normal drag coefficient $C_{\mathrm{Dn}l}$, so the drag force contribution from the exposed line is \begin{equation} \label{fDxel} f_{\mathrm{Dx}el} = \frac12 \rho\, v^2 C_{\mathrm{Dn}l} \frac{d_{\mathrm{n}l} (s_f-l_f)}{s_f} \end{equation} in which the final term is the normal drag area of the exposed part of the line.
Substituting from expressions (\ref{fDxf}) and (\ref{fDxel}) into (\ref{fDx2}), and equating (\ref{fDx1}) and (\ref{fDx2}), we arrive at the final formula for the $x$-drag coefficient of the derived line type \begin{equation} \label{CDx} \C{Dx} = C_{\mathrm{Dn}f} \frac{d_f l_f}{d_{\mathrm{n}l} s_f} + C_{\mathrm{Dn}l} \frac{(s_f-l_f)}{s_f} \end{equation} Now, since both the base line type and the floats are isotropic, so is the derived line type – the right-hand side of equation (\ref{CDx}) is the same for all flow directions in the plane normal to the line axis – so the wizard sets \begin{equation} \C{Dy} =\ \sim \end{equation}
If the base line type is of the general category, then the normal drag coefficient is defined independently in the $x$ and $y$ directions as $C_{\mathrm{Dx}l}$ and $C_{\mathrm{Dy}l}$ respectively. Repeating the above analysis in this case gives \begin{equation} \C{Dx} = C_{\mathrm{Dn}f} \frac{d_f l_f}{d_{\mathrm{n}l} s_f} + C_{\mathrm{Dx}l} \frac{(s_f-l_f)}{s_f} \\ \end{equation} Again, if the base line type drag coefficient is isotropic (i.e. if $C_{\mathrm{Dy}l}{=}$'~'), then so must be the drag coefficient of the derived line type, and the wizard sets \begin{equation} \C{Dy} =\ \sim \end{equation} If, however, the base line type value $C_{\mathrm{Dy}l}{\neq}$'~', but instead has a numerical value, then the wizard applies the formula \begin{equation} \C{Dy} = C_{\mathrm{Dn}f} \frac{d_f l_f}{d_{\mathrm{n}l} s_f} + C_{\mathrm{Dy}l} \frac{(s_f-l_f)}{s_f} \end{equation} The normal drag coefficient for the float, $C_{\mathrm{Dn}f}$, is always isotropic, so has the same value in all normal directions.
If the base line type (of either category) has drag which varies with Reynolds number, then another variable data set is created for the drag variation with Reynolds number for the derived line type. The appropriate formula for $\C{Dx}$ and $\C{Dy}$ are applied to each of the drag coefficients defining the base variable data to populate this derived variable data set.