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
Proceedings of the 8th ITTC, Madrid, 1957
Grigson CWB (1999) A planar friction algorithm and its use in analysing hull resistance. Trans RINA, pp 76–115
Katsui T, Asai H, Himeno Y, Tahara Y (2005) The proposal of a new friction line. Fifth Osaka colloquium on advanced CFD applications to ship flow and hull form design, Osaka, Japan
Coles D (1956) The law of the wake in the turbulent boundary layer. J Fluid Mech 1:191–226
Eça L, Hoekstra M (2005) On the accuracy of the numerical prediction of scale effects on ship viscous resistance—computational methods in marine engineering, CIMNE Barcelona, international conference on computational methods in marine engineering (MARINE 2005), Oslo, Norway
Menter FR (1997) Eddy viscosity transport equations and their relation to the kε model. J Fluid Eng 119:876–884
Spalart PR, Allmaras SR (1992) A oneequation turbulence model for aerodynamic flows. AIAA 30th aerospace sciences meeting, Reno
Wilcox DC (1998) Turbulence modeling for CFD, Second edn. DCW Industries, La Canada
Kok JC (1999) Resolving the dependence on freestream values for the kω turbulence model. NLRTP99295
Menter FR (1994) Twoequation Eddyviscosity turbulence models for engineering applications. AIAA J 32:1598–1605
Menter FR, Egorov Y, Rusch D (2006) Steady and unsteady flow modelling using the \( k \text{}\!\sqrt k L \) model. Turbulence, Heat and Mass Transfer 5 Proceeding of The International Symposium on Turbulence, Heat and Mass Transfer—Dubrovnik, Croatia, September 2529, 2006, vol 1. Begell, Redding, pp 403–406
José MQB, Jacob Eça L (2000) 2D incompressible steady flow calculations with a fully coupled method. VI Congresso Nacional de Mecânica Aplicada e Computacional, Aveiro
Eça L, Hoekstra M (2004) On the grid sensitivity of the wall boundary condition of the kω model. J Fluid Eng 126(6):900–910
ERCOFTAC Classic Collection Database—http://cfd.me.umist.ac.uk/ercoftac
Pecnik R, Sanz W, Gehrer A, Woisetschläger (2003) Transition modeling using two different intermittency transport equations. Flow Turbul Combust 70:299–323
Menter FR, Esch T, Kubacki S (2002) Transition modeling based on local variables. Proceedings of the 5th international symposium on engineering turbulence modelling and measurements, Mallorca, Spain, pp 555–564
Eça L, Hoekstra M (2007) The numerical friction line. EU project the virtual towing tank, work package 1, D.1.1.2
Vinokur M (1983) On onedimensional stretching functions for finitedifference calculations. J Comput Phys 50:215–234
Roache PJ (1998) Verification and validation in computational science and engineering. Hermosa Publishers, Albuquerque, New Mexico
Eça L, Hoekstra M (2006) On the influence of the iterative error in the numerical uncertainty of CFD predictions. 26th Symposium on Naval Hydrodynamics, Rome, Italy
Eça L, Hoekstra M (2005) On the influence of grid topology on the accuracy of ship viscous flow calculations. Fifth Osaka colloquium on advanced CFD applications to ship flow and hull form design, Osaka, Japan
White FM (1991) Viscous fluid flow, 2nd edn. McGrawHill, New York
Schlichting H (1979) Boundary layer theory, 7th edn. McGrawHill, New York
Schoenherr Karl E (1932) Resistance of flat surfaces. Trans SNAME 40:279–313
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
Keywords
 Friction resistance coefficient
 Eddyviscosity models
 Iterative error
 Discretization error