Skip to main content

The numerical friction line


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 eddy-viscosity models have been selected: the one-equation turbulence models of Menter and Spalart–Allmaras; the k-ω two-equation model proposed by Wilcox and its TNT, BSL and SST variants and the \( k \text{-}\!\sqrt k L \) two-equation 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 round-off 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 laminar-to-turbulent 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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12



Reynolds-averaged Navier–Stokes equations






Shear-stress transport


International Towing Tank Conference


  1. Proceedings of the 8th ITTC, Madrid, 1957

  2. Grigson CWB (1999) A planar friction algorithm and its use in analysing hull resistance. Trans RINA, pp 76–115

  3. 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

  4. Coles D (1956) The law of the wake in the turbulent boundary layer. J Fluid Mech 1:191–226

    MATH  Article  MathSciNet  Google Scholar 

  5. 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

  6. Menter FR (1997) Eddy viscosity transport equations and their relation to the k-ε model. J Fluid Eng 119:876–884

    Article  Google Scholar 

  7. Spalart PR, Allmaras SR (1992) A one-equation turbulence model for aerodynamic flows. AIAA 30th aerospace sciences meeting, Reno

  8. Wilcox DC (1998) Turbulence modeling for CFD, Second edn. DCW Industries, La Canada

    Google Scholar 

  9. Kok JC (1999) Resolving the dependence on free-stream values for the k-ω turbulence model. NLR-TP-99295

  10. Menter FR (1994) Two-equation Eddy-viscosity turbulence models for engineering applications. AIAA J 32:1598–1605

    Article  Google Scholar 

  11. 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 25-29, 2006, vol 1. Begell, Redding, pp 403–406

  12. José MQB, Jacob Eça L (2000) 2-D incompressible steady flow calculations with a fully coupled method. VI Congresso Nacional de Mecânica Aplicada e Computacional, Aveiro

  13. Eça L, Hoekstra M (2004) On the grid sensitivity of the wall boundary condition of the k-ω model. J Fluid Eng 126(6):900–910

    Article  Google Scholar 

  14. ERCOFTAC Classic Collection Database—

  15. Pecnik R, Sanz W, Gehrer A, Woisetschläger (2003) Transition modeling using two different intermittency transport equations. Flow Turbul Combust 70:299–323

  16. 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

  17. Eça L, Hoekstra M (2007) The numerical friction line. EU project the virtual towing tank, work package 1, D.1.1.2

  18. Vinokur M (1983) On one-dimensional stretching functions for finite-difference calculations. J Comput Phys 50:215–234

    MATH  Article  MathSciNet  Google Scholar 

  19. Roache PJ (1998) Verification and validation in computational science and engineering. Hermosa Publishers, Albuquerque, New Mexico

  20. 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

  21. 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

  22. White FM (1991) Viscous fluid flow, 2nd edn. McGraw-Hill, New York

  23. Schlichting H (1979) Boundary layer theory, 7th edn. McGraw-Hill, New York

  24. Schoenherr Karl E (1932) Resistance of flat surfaces. Trans SNAME 40:279–313

    Google Scholar 

Download references


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.

Author information

Authors and Affiliations


Corresponding author

Correspondence to L. Eça.

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: δRE and ΔM. δRE is the error estimation obtained by Richardson extrapolation

$$ \phi _{i} - \phi _{o} = \delta _{{{\text{RE}}}} = \alpha h_{i}^{p} $$

and ΔM is the data range:

$$ \Updelta _{\rm M} = \max (|\phi _{j} - \phi _{i} |)\quad 1 \le i,j \le n_{g} , $$

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. 1.

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

  2. 2.

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

  3. 3.

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

  4. 4.

    Otherwise → Oscillatory convergence.

We can summarize our procedure for the estimation of the numerical uncertainty, valid for a nominally second-order accurate method, as follows:

  1. 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.

  2. 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 , 1.25ΔM)

    • For p ≥ 2.05: \( U_\phi \)  = max(1.25 δ * RE  + U s , 1.25ΔM)

    • If monotonic convergence is not observed: \( U_\phi \) = 3 ΔM

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 round-off errors are negligible.

About this article

Cite this article

Eça, L., Hoekstra, M. The numerical friction line. J Mar Sci Technol 13, 328–345 (2008).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Friction resistance coefficient
  • Eddy-viscosity models
  • Iterative error
  • Discretization error