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 $\alpha \ = \ - 1.43$.

中文翻译：

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对数美学曲线及其与流线的流体流动关系☆

摘要

对数美学曲线（LAC）是具有线性对数曲率图（LCG）的一系列美学曲线。它包含众所周知的美学曲线，例如回旋曲线，对数螺旋和圆渐开线。LAC在美学设计中一直发挥着举足轻重的作用。但是，其在功能设计中的应用是一个未知领域，例如，LAC与流体流动模式之间的关系可能有助于设计更好的船体和防波堤。我们通过以稳定状态下的流线阐明LAC与流模式之间的关系来解决此问题。我们通过Euler方程讨论了在压力梯度影响下LAC路径线是如何形成的，以及在特殊情况下LAC流线是如何形成的。得出隐式和显式函数的LCG梯度（$ \ alpha $），证明了当三次函数的导数为零时，显式函数拐点处的LCG梯度始终为0。由于LAC参数表示的复杂性，几乎不可能得出LAC流线的一般表示。我们通过分析由周围各种形状的LAC生成的类似翼型障碍物的不可压缩流形成的流线来解决这一问题，$ {\ 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} \} $，并使用FreeFem ++模拟流线达到稳定状态。我们发现，所得流线的LCG梯度接近回旋曲线。当障碍物形状与圆形几乎相同时（$ \ alpha \ = \ 20 $），尽管接近稳态，但与障碍物相邻的流线具有许多曲率极值。流速变化对于$ \ alpha \ = \-1.43 $是最低的，并且随着$ \ alpha $从$ \ alpha \ = \-1.43 $增大或减小而变大。