Late time evolution of unforced inviscid two-dimensional turbulence
Late time evolution of unforced inviscid two-dimensional turbulence
复制标题
非受迫无粘性二维湍流的晚期演化
DOI：
10.1017/s0022112009991121
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发表时间：
2009
影响因子：
3.7
通讯作者：
C. Tran
中科院分区：
文献类型：
--
作者：
D. Dritschel;R. K. Scott;C. Macaskill;G. Gottwald;C. Tran
我们提出了一个新的统一模型，用于无粘极限下自由衰减二维湍流的小、中和大规模演化。新模型的核心是最近的涡旋自相似理论（Dritschel et al., Phys. Rev. Lett., vol. 101, 2008, no. 094501），适用于不断扩大的涡旋群体所跨越的中间尺度范围。预计该范围具有陡峭的 k−5 能谱。在小尺度上，这让位于巴彻勒 (Batchelor, Phys. Fluids, vol. 12, 1969, p. 233) k−3 能谱，对应于（前向）熵（均方涡度）级联，或者物理上对应于涡旋碰撞产生的细化丝状碎片。这个小范围的范围几乎携带着所有的熵，但能量可以忽略不计。在大尺度上，最大涡旋尺寸（半径约为 t1/6）的缓慢增长意味着相应的缓慢的逆能量级联。我们认为，这种极其缓慢的增长使得大尺度接近均分（Kraichnan, Phys. Fluids, vol. 10, 1967, p. 1417; Fox & Orszag, Phys. Fluids, vol. 12, 1973, p. 169），最终导致那里的 k1 能谱。总而言之，我们提出的模型在大尺度上具有能谱 ℰ(k, t) ∝ t1/3k1，在涡流总体上具有 ℰ(k, t) ∝ t−2/3k−5 ，最后在以不相干丝状碎片为主的呈指数扩大的小尺度范围内具有 ℰ(k, t) ∝ t−1k−3 。对我们模型的支持分为两部分。首先，我们使用新颖的高分辨率胞内涡模拟来解决大尺度和超大规模（比任何涡旋大得多）的演化。这验证了均分，但更重要的是让我们更好地理解均分的方法。其次，我们通过一组特别高分辨率的直接数值模拟来解决中尺度和小尺度问题。
We propose a new unified model for the small, intermediate and large-scale evolution of freely decaying two-dimensional turbulence in the inviscid limit. The new model's centerpiece is a recent theory of vortex self-similarity (Dritschel et al., Phys. Rev. Lett., vol. 101, 2008, no. 094501), applicable to the intermediate range of scales spanned by an expanding population of vortices. This range is predicted to have a steep k−5 energy spectrum. At small scales, this gives way to Batchelor's (Batchelor, Phys. Fluids, vol. 12, 1969, p. 233) k−3 energy spectrum, corresponding to the (forward) enstrophy (mean square vorticity) cascade or, physically, to thinning filamentary debris produced by vortex collisions. This small-scale range carries with it nearly all of the enstrophy but negligible energy. At large scales, the slow growth of the maximum vortex size (~t1/6 in radius) implies a correspondingly slow inverse energy cascade. We argue that this exceedingly slow growth allows the large scales to approach equipartition (Kraichnan, Phys. Fluids, vol. 10, 1967, p. 1417; Fox & Orszag, Phys. Fluids, vol. 12, 1973, p. 169), ultimately leading to a k1 energy spectrum there. Put together, our proposed model has an energy spectrum ℰ(k, t) ∝ t1/3k1 at large scales, together with ℰ(k, t) ∝ t−2/3k−5 over the vortex population, and finally ℰ(k, t) ∝ t−1k−3 over an exponentially widening small-scale range dominated by incoherent filamentary debris. Support for our model is provided in two parts. First, we address the evolution of large and ultra-large scales (much greater than any vortex) using a novel high-resolution vortex-in-cell simulation. This verifies equipartition, but more importantly allows us to better understand the approach to equipartition. Second, we address the intermediate and small scales by an ensemble of especially high-resolution direct numerical simulations.