Log-aesthetic curves and their relation to fluid flow patterns in terms of streamlines☆

Abstract

The log-aesthetic curve (LAC) is a family of aesthetic curves with linear logarithmic curvature graphs (LCGs). It encompasses well-known aesthetic curves such as clothoid, logarithmic spiral, and circle involute. LAC has been playing a pivotal role in aesthetic design. However, its application for functional design is an uncharted territory, e.g. the relationship between LAC and fluid flow patterns may aid in designing better ship hulls and breakwaters. We address this problem by elucidating the relationship between LAC and flow patterns in terms of streamlines at a steady state. We discussed how LAC pathlines form under the influence of pressure gradient via Euler's equation and how LAC streamlines are formed in a special case. LCG gradient ($\alpha $) for implicit and explicit functions is derived, and it is proven that the LCG gradient at the inflection points of explicit functions is always 0 when its third derivative is nonzero. Due to the complexity of the parametric representation of LAC, it is almost impossible to derive the general representation of LAC streamlines. We address this by analyzing the streamlines formed by incompressible flow around an airfoil-like obstacle generated with LAC having various shapes, ${\alpha _r} = \ \{ { - 20,{\rm{\ }} - 5,{\rm{\ }} - 1,{\rm{\ }} - 0.5,{\rm{\ }} - 0.15,{\rm{\ }}0,{\rm{\ }}1,{\rm{\ }}2,{\rm{\ }}3,{\rm{\ }}4,{\rm{\ }}20} \}$, and simulating the streamlines using FreeFem++ reaching a steady state. We found that the LCG gradient of the resultant streamlines is close to that of a clothoid. When the obstacle shape is almost the same as that of a circle ($\alpha \ = \ 20$), the streamlines adjacent to the obstacles have numerous curvature extrema despite nearing steady state. The flow speed variation is the lowest for $\alpha \ = \ - 1.43$ and gets higher as $\alpha$ is increased or decreased from $\alpha \ = \ - 1.43$.