Abstract
This paper presents a study on the numerical calculation of the friction resistance coefficient of an infinitely thin plate as a function of the Reynolds number. Seven eddyviscosity models have been selected: the oneequation turbulence models of Menter and Spalart–Allmaras; the kω twoequation model proposed by Wilcox and its TNT, BSL and SST variants and the \( k \text{}\!\sqrt k L \) twoequation model. The flow has been computed at 14 Reynolds numbers in sets of seven geometrically similar Cartesian grids to allow a reliable estimation of the numerical uncertainty. The effect of the computational domain size has been reduced to negligible levels (below the numerical uncertainty). And the same holds for the iterative and roundoff errors. In the finest grids of each set, the numerical uncertainty of the friction resistance coefficient is always below 1%. Special attention has further been given to the solution behaviour in the laminartoturbulent transition region. Curve fits have been applied to the data obtained at the 14 Reynolds numbers and the numerical friction lines are compared with four proposals from the open literature: the 1957 ITTC line, the Schoenherr line and the lines suggested by Grigson and Katsui et al. The differences between the numerical friction lines obtained with the seven turbulence models are smaller than the differences between the four lines proposed in the open literature.
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Abbreviations
 RANS:

Reynoldsaveraged Navier–Stokes equations
 TNT:

Turbulent/nonturbulent
 BSL:

Baseline
 SST:

Shearstress transport
 ITTC:

International Towing Tank Conference
References
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Acknowledgments
The present work has been carried out in the VIRTUE project, an Integrated Project in the sixth Framework Programme “Sustainable development, global change and ecosystems” under grant 516201 from the European Commission. This support is gratefully acknowledged.
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Appendix 1
Appendix 1
The procedure for uncertainty estimation is based on a least squares version of the Grid Convergence Index method proposed by Roache [19]. It uses two error estimators: δ_{RE} and Δ_{M}. δ_{RE} is the error estimation obtained by Richardson extrapolation
and Δ_{M} is the data range:
where \( \phi_i \) is the numerical solution of any local or integral scalar quantity on a given grid (designated by the subscript _{ i }), \( \phi_o \) is the estimated exact solution, α is a constant, h _{ i } is a parameter which identifies the representative grid cell size, p is the observed order of accuracy and n _{ g } is the number of grids available. \( \phi_o \), α and p are obtained with a least squares fit of the data. When more than three grids are available and the least squares root approach is applied, it is not easy to classify the apparent convergence condition because the data may exhibit scatter. First, we establish the apparent order of convergence p from the least squares fit. Next, to identify the cases of oscillatory convergence or divergence, we also determine p* using \( {\phi_i^*}=\phi_{i+1} \phi_i \); this fit includes only n _{ g }−1 differences. The apparent convergence condition is then decided as follows:

1.
p > 0 for \( \phi \) → Monotonic convergence.

2.
p < 0 for \( \phi \) → Monotonic divergence.

3.
p* < 0 for \( \phi^* \) → Oscillatory divergence.

4.
Otherwise → Oscillatory convergence.
We can summarize our procedure for the estimation of the numerical uncertainty, valid for a nominally secondorder accurate method, as follows:

•
The observed order of accuracy is estimated with the least squares root technique to identify the apparent convergence condition according to the definition given above.

•
Monotonic convergence:

For 0.95 ≤ p < 2.05: \( U_\phi \) = 1.25 δ _{RE} + U _{ s }

For 0 < p < 0.95: \( U_\phi \) = min(1.25 δ _{RE} + U _{ s } , 1.25Δ_{M})

For p ≥ 2.05: \( U_\phi \) = max(1.25 δ ^{*} _{RE} + U _{ s } , 1.25Δ_{M})

If monotonic convergence is not observed: \( U_\phi \) = 3 Δ_{M}

U _{ s } stands for the standard deviation of the fit and δ ^{*}_{RE} is obtained with Richardson extrapolation using p equal to the theoretical value to obtain the error estimator, δ ^{*}_{RE} .
In the previous description, we have assumed that the iterative and roundoff errors are negligible.
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Eça, L., Hoekstra, M. The numerical friction line. J Mar Sci Technol 13, 328–345 (2008). https://doi.org/10.1007/s0077300800181
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DOI: https://doi.org/10.1007/s0077300800181