jats:pA direct numerical simulation (DNS) database of freely propagating statistically planar turbulent premixed flames with a range of different turbulent Reynolds numbers has been used to assess the performance of algebraic flame surface density (FSD) models based on a fractal representation of the flame wrinkling factor. The turbulent Reynolds number Rejats:subt</jats:sub>has been varied by modifying the Karlovitz number Ka and the Damköhler number Da independently of each other in such a way that the flames remain within the thin reaction zones regime. It has been found that the turbulent Reynolds number and the Karlovitz number both have a significant influence on the fractal dimension, which is found to increase with increasing Rejats:subt</jats:sub>and Ka before reaching an asymptotic value for large values of Rejats:subt</jats:sub>and Ka. A parameterisation of the fractal dimension is presented in which the effects of the Reynolds and the Karlovitz numbers are explicitly taken into account. By contrast, the inner cut-off scale normalised by the Zel’dovich flame thickness<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1">mml:msubmml:mrowmml:miη</mml:mi></mml:mrow>mml:mrowmml:mii</mml:mi></mml:mrow></mml:msub>mml:mo/</mml:mo>mml:msubmml:mrowmml:miδ</mml:mi></mml:mrow>mml:mrowmml:miz</mml:mi></mml:mrow></mml:msub></mml:math>does not exhibit any significant dependence on Rejats:subt</jats:sub>for the cases considered here. The performance of several algebraic FSD models has been assessed based on various criteria. Most of the algebraic models show a deterioration in performance with increasing the LES filter width.</jats:p>