An important idea underlying a plausible dynamical theory of circulation in three-dimensional turbulence is the so-called area rule, according to which the probability density function (PDF) of the circulation around closed loops depends only on the minimal area of the loop, not its shape. We assess the robustness of the area rule, for both planar and nonplanar loops, using high-resolution data from direct numerical simulations. For planar loops, the circulation moments for rectangular shapes match those for the square with only small differences, these differences being larger when the aspect ratio is farther from unity and when the moment order increases. The differences do not exceed about 5% for any condition examined here. The aspect ratio dependence observed for the second-order moment is indistinguishable from results for the Gaussian random field (GRF) with the same two-point correlation function (for which the results are order-independent by construction). When normalized by the SD of the PDF, the aspect ratio dependence is even smaller ( < 2%) but does not vanish unlike for the GRF. We obtain circulation statistics around minimal area loops in three dimensions and compare them to those of a planar loop circumscribing equivalent areas, and we find that circulation statistics match in the two cases only when normalized by an internal variable such as the SD. This work highlights the hitherto unknown connection between minimal surfaces and turbulence.

三维湍流中环流的合理动力学理论背后的一个重要思想是所谓的面积规则，根据该规则，闭环周围环流的概率密度函数 (PDF) 仅取决于环的最小面积，而不取决于它的形状。我们使用来自直接数值模拟的高分辨率数据评估平面和非平面环的面积规则的稳健性。对于平面环，矩形的环流力矩与正方形的环流力矩相匹配，只有很小的差异，当纵横比远离统一并且力矩阶数增加时，这些差异会更大。对于此处检查的任何条件，差异不超过约 5%。观察到的二阶矩的纵横比依赖性与具有相同两点相关函数的高斯随机场 (GRF) 的结果无法区分（其结果与构造无关）。当通过 PDF 的 SD 进行归一化时，纵横比依赖性甚至更小（< 2%），但不会像 GRF 那样消失。我们在三个维度上获得了围绕最小面积环的环流统计数据，并将它们与包围等效区域的平面环的环流统计数据进行比较，我们发现只有当通过 SD 等内部变量进行归一化时，环流统计数据才在这两种情况下匹配。这项工作突出了最小表面和湍流之间迄今为止未知的联系。