Institut für Theoretische Physik I, Ruhr-Universität Bochum, Universitätsstraße 150, 44780 Bochum, Germany.
Department of Physics and INFN, University of Rome "Tor Vergata," Via della Ricerca Scientifica 1, 00133 Roma, Italy.
Phys Rev E. 2018 Aug;98(2-1):023104. doi: 10.1103/PhysRevE.98.023104.
We compare different approaches towards an effective description of multiscale velocity field correlations in turbulence. Predictions made by the operator-product expansion, the so-called fusion rules, are placed in juxtaposition to an approach that interprets the turbulent energy cascade in terms of a Markov process of velocity increments in scale. We explicitly show that the fusion rules are a direct consequence of the Markov property provided that the structure functions exhibit scaling in the inertial range. Furthermore, the limit case of joint velocity gradient and velocity increment statistics is discussed and put into the context of the notion of dissipative anomaly. We generalize a prediction made by the multifractal model derived by Benzi et al. [R. Benzi et al., Phys. Rev. Lett. 80, 3244 (1998)PRLTAO0031-900710.1103/PhysRevLett.80.3244] to correlations among inertial range velocity increment and velocity gradients of any order. We show that for the case of squared velocity gradients such a relation can be derived from first principles in the case of Burgers equations. Our results are benchmarked by intensive direct numerical simulations of Burgers turbulence.
我们比较了几种方法，以有效地描述湍流中的多尺度速度场相关性。将算子乘积展开（所谓的融合规则）的预测与一种方法进行了对比，该方法根据速度增量的马尔可夫过程来解释湍流能量级联。我们明确表明，只要结构函数在惯性范围内具有标度性，融合规则就是马尔可夫性质的直接结果。此外，还讨论了联合速度梯度和速度增量统计的极限情况，并将其置于耗散异常的概念背景下。我们推广了 Benzi 等人提出的多重分形模型的一个预测[R. Benzi 等人，物理评论快报 80, 3244 (1998)PRLTAO0031-900710.1103/PhysRevLett.80.3244]，使其适用于任何阶次的惯性区速度增量和速度梯度之间的相关性。我们表明，在 Burgers 方程的情况下，对于平方速度梯度的情况，可以从第一性原理推导出这样的关系。我们的结果通过 Burgers 湍流的密集直接数值模拟进行了基准测试。
