Tran Chuong V, Bowman John C
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1.
Phys Rev E Stat Nonlin Soft Matter Phys. 2004 Mar;69(3 Pt 2):036303. doi: 10.1103/PhysRevE.69.036303. Epub 2004 Mar 23.
We study quasisteady inverse cascades in unbounded and bounded two-dimensional turbulence driven by time-independent injection and dissipated by molecular viscosity. It is shown that an inverse cascade that carries only a fraction r of the energy input to the largest scales requires the enstrophy-range energy spectrum to be steeper than k(-5) (ruling out a direct cascade) unless 1-r<<1. A direct cascade requires the presence of an inverse cascade that carries virtually all energy to the largest scales (1-r<<1). These facts underlie the robustness of the Kolmogorov-Kraichnan k(-5/3) inverse cascade, which is readily observable in numerical simulations without an accompanying direct enstrophy cascade. We numerically demonstrate an instance where the k(-5/3) inverse-cascading range is realizable with 79% of the energy injection dissipated within the energy range and virtually all of the enstrophy dissipated in the vicinity of the forcing region. As equilibrium is approached, the respective logarithmic slopes -alpha and -beta of the ranges of wave numbers lower and higher than the forcing wave number satisfy alpha+beta approximately 8. These results are consistent with recent theoretical predictions.
我们研究了由与时间无关的注入驱动并由分子粘性耗散的无界和有界二维湍流中的准稳态逆级联。结果表明，仅将能量输入的一小部分r输送到最大尺度的逆级联要求涡量范围的能谱比k^(-5)更陡（排除直接级联），除非1 - r << 1。直接级联需要存在一个将几乎所有能量输送到最大尺度的逆级联（1 - r << 1）。这些事实构成了Kolmogorov - Kraichnan k^(-5/3)逆级联稳健性的基础，在数值模拟中很容易观察到这种逆级联，而没有伴随的直接涡量级联。我们通过数值证明了一个实例，其中k^(-5/3)逆级联范围是可实现的，79%的能量注入在能量范围内耗散，几乎所有的涡量在强迫区域附近耗散。当接近平衡时，低于和高于强迫波数的波数范围各自的对数斜率 -α和 -β满足α + β ≈ 8。这些结果与最近的理论预测一致。
