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# Extrapolation to full scale

This section describes the method used for extrapolation of skin friction to full scale. Implementing or understanding the method is not required for using the database, as the entire method is coded and runs behind the interface. This page is only for background information for the method used.

For extrapolation to full scale Reynolds number the Granville similarity laws are used. The reason for choosing this particular extrapolation have several reasons. First of all it is based on the velocity shift of the boundary seen when fluid traverses a rough surface. It is therefore possible to derive specific equations for several different types of flows (flat plate, pipe flow or coquette cell for instance), which makes it more valid to compare lab measurements from different types of measurement devices for rough measurements. It is also very suitable for indirect methods (not measuring the local skin friction by for example shear stress measurement of boundary layer profile measurements) using only a towing force for flat plate or torque for couette cell. Effects of displacement thickness for a developing boundary is also included in the equations. Finally, it is the recommended method from [ITTC, 2017]. For the full explanation of the Granville similarity see [Granville, 1987] or [Demirel 2014]. Expressed as

(1)

(2)

where k is the roughness height, L is the plate length, is the roughness function and ‘ is the slope of the roughness function and finally R and S subscripts refers to rough and smooth surface respectively. The above equations are solved iteratively based on the measured data for the plate until the slope of as a function of calculated numerically based on the measurement points satisfies equation 2 (by least squares interpolation). For the smooth surface the Schoenherr smooth skin friction line is used (which is also solved iteratively)

(3)

The same value of must be used in equation 1. Therefore the smooth value is not the value of equation 3 for the same as the measured rough point, but is the value of using on the right hand side of equation 3.

Second step in the method is to plot as a function of and based on the value of from equation 2 at one measured point for a rough surface to shift the smooth skin friction curve a distance of in the direction, see Figure 16. This line corresponds to C_{F} derived from the measurement point to a plate with varying length, but same speed as the measurement.

Schoenherr smooth skin friction and shift. C_{FR} is the measured data.

The next step is to calculate the curve of constant using equation 4, where the constant value of is defined by a point from the measurements

(4)

Thus using the constant value of for the measurement point the curve of constant can be added by solving for as a function of . The final step is to shift the curve a distance of , as can be seen in the fubure below. The intersection between the shifted Schoenherr and constant is the skin friction at a plate corresponding to a length of at a velocity corresponding to the velocity for the measurement point.

Constant L^{+} and shift.

This extrapolation in the plate length direction usually carries the estimation of C_{F }almost to full scale value in terms of Re_{L} for a vessel. To extrapolate in the velocity direction this method cannot be used directly, insteada roughness functions must be used if the measured velocities does not match that of the full scale ship. In the present study the Grigson formulation is used (see Grigson(1992))

where the velocity shift also can be expressed as

=

Whereas the extrapolation to full scale length scale is independent of the choice of roughness height k, the extrapolation along the velocity direction is not. Therefore the choice of k must be fitted to the measured data, which will be done by introducing an efficiency parameter C reformulating the above equation to

which will also allow for some scaling of topologically similar surfaces (surfaces with different roughness but same efficiency parameter for instance barnacle surface with different size of barnacles). With the model tests all variables in the last equation are known except and C, the parameter for can be extracted using the experimental data, allowing a rough skin friction curve to be extrapolated also in the velocity dimension by increasing the velocity U and keeping constant. The Grigson roughness function in the fully rough regime increases as a function of Re which is not physically correct. Therefore the formulation above is modified such that if the slope of is positive then

As an example the full extrapolation procedure result is shown graphically in the plot below. It shows two rough antifouling (65 and 110mikron AHR) measurement with a plate/ship length of 219m. Dotted lines are rough measurements at plate length (blue and black) and at ship length (orange and green). The same colors are the Grigson velocity extrapolation based on the measurements at plate and ship length. The vertical green line is the target Reynolds number for the vessel and the grey line is the hydraulically smooth Schoenherr line. The resulting used in the fuel increase tool in the home page is the difference between the two blue points at the target Reynolds number.

Example of extrapolation

For the interactive database the entire procedure above is updated whenever the button Calculate Friction is pushed. Behind the visual interface this is accomplished by executing a Fortran routine having access to the measured rough skin friction measurements and the data needed from the user to update the extrapolation (Ship length, Speed and Efficiency parameter). The Fortran routine uses a combination least square curve fitting, intersection of curves and where the computational requirements are relatively high interpolation in tables with tabulated data to keep the update as instantaneous as possible. The update time is still a couple of seconds.