Blackbourn Luke A K, Tran Chuong V
School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, United Kingdom.
Phys Rev E Stat Nonlin Soft Matter Phys. 2011 Oct;84(4 Pt 2):046322. doi: 10.1103/PhysRevE.84.046322. Epub 2011 Oct 28.
Large-scale dissipation mechanisms have been routinely employed in numerical simulations of two-dimensional turbulence to absorb energy at large scales, presumably mimicking the quasisteady picture of Kraichnan in an unbounded fluid. Here, "side effects" of such a mechanism--mechanical friction--on the small-scale dynamics of forced two-dimensional Navier-Stokes turbulence are elaborated by both theoretical and numerical analysis. Given a positive friction coefficient α, viscous dissipation of enstrophy has been known to vanish in the inviscid limit ν→0. This effectively renders the scale-neutral friction the only mechanism responsible for enstrophy dissipation in that limit. The resulting dynamical picture is that the classical enstrophy inertial range becomes a dissipation range in which the dissipation of enstrophy by friction mainly occurs. For each α>0, there exists a critical viscosity ν(c), which depends on physical parameters, separating the regimes of predominant viscous and frictional dissipation of enstrophy. It is found that ν(c)=[η'(1/3)/(Ck(f)(2))]exp[-η'(1/3)/(Cα)], where η' is half the enstrophy injection rate, k(f) is the forcing wave number, and C is a nondimensional constant (the Kraichnan-Batchelor constant). The present results have important theoretical and practical implications. Apparently, mechanical friction is a poor choice in numerical attempts to address fundamental issues concerning the direct enstrophy transfer in two-dimensional Navier-Stokes turbulence. Furthermore, as relatively strong friction naturally occurs on the surfaces and at lateral boundaries of experimental fluids as well as at the interfaces of shallow layers in geophysical fluid models, the frictional effects discussed in this study are crucial in understanding the dynamics of these systems.
在二维湍流的数值模拟中，大规模耗散机制已被常规用于在大尺度上吸收能量，大概是模拟无界流体中克莱奇南的准稳态图景。在此，通过理论和数值分析详细阐述了这种机制——机械摩擦——对强迫二维纳维 - 斯托克斯湍流小尺度动力学的“副作用”。给定正的摩擦系数α，已知在无粘极限ν→0时涡量的粘性耗散会消失。这有效地使尺度中性摩擦成为该极限下负责涡量耗散的唯一机制。由此产生的动力学图景是，经典的涡量惯性范围变成了一个耗散范围，其中涡量主要通过摩擦耗散。对于每个α>0，存在一个临界粘性ν(c)，它取决于物理参数，将涡量主要由粘性耗散和摩擦耗散的区域分开。发现ν(c)=[η'(1/3)/(Ck(f)(2))]exp[-η'(1/3)/(Cα)]，其中η'是涡量注入率的一半，k(f)是强迫波数，C是一个无量纲常数（克莱奇南 - 巴彻勒常数）。目前的结果具有重要的理论和实际意义。显然，在数值研究中试图解决二维纳维 - 斯托克斯湍流中直接涡量传递的基本问题时，机械摩擦不是一个好的选择。此外，由于在实验流体的表面和横向边界以及地球物理流体模型中浅层的界面处自然会出现相对较强的摩擦，本研究中讨论的摩擦效应对于理解这些系统的动力学至关重要。
