The numerical friction line

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 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.

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

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

and Δ_M 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. 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 second-order 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 round-off errors are negligible.