Exact two-dimensionalization of rapidly rotating large-Reynolds-number flows
Exact two-dimensionalization of rapidly rotating large-Reynolds-number flows
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快速旋转大雷诺数流的精确二维化
DOI：
10.1017/jfm.2015.569
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发表时间：
2015
影响因子：
3.7
通讯作者：
B. Gallet
中科院分区：
文献类型：
--
作者：
B. Gallet
我们考虑牛顿流体在三维区域中的流动，绕垂直轴旋转，并由垂直不变的水平体力驱动。该系统承认垂直不变的解决方案，满足二维Navier-Stokes方程。在高雷诺数和没有整体旋转，这样的解决方案通常是不稳定的三维扰动。相比之下，足够强的全球旋转，我们严格证明，二维（可能是动荡的）解决方案是稳定的垂直依赖扰动。我们首先考虑三维旋转Navier-Stokes方程线性化周围的统计稳定的二维流动解决方案。我们表明，当全局旋转足够快时，这种基流对垂直相关扰动是线性稳定的：在Rossby数依赖于Rossby数的阈值$Ro_{c}（Re）$下，只要初始的3D扰动很小，基流在长时间极限内完全变为2D。我们称这种性质为线性二维化。我们计算$Ro_{c}（Re）$上的显式下界，从而确定参数空间$（Re，Ro）$中发生这种精确二维化的区域。我们用强迫强度代替均方根速度给出了类似的结果：当强迫Rossby数Ro^{（f）}$低于Grashoff数相关阈值Ro_{c}^{（f）}（Gr）$时，二维Navier-Stokes方程的全局吸引子对垂直相关扰动是线性稳定的.然后，我们考虑完全非线性的三维旋转Navier-Stokes方程，并证明绝对二维化：我们表明，低于某个阈值$Ro_{\mathit{abs}}^{（f）}（Gr）$的强迫为基础的Rossby数，流成为二维的长时间限制，无论初始条件（包括初始三维扰动的任意大幅度）。这些结果揭示了一些旋转湍流的几个基本问题：对于任意雷诺数Re$和足够小的Rossby数，该系统被吸引到纯二维流动的解决方案，显示没有能量耗散异常，没有气旋反气旋不对称。最后，这些结果的适用性的波动湍流理论来描述定常旋转湍流有界域的挑战。
We consider the flow of a Newtonian fluid in a three-dimensional domain, rotating about a vertical axis and driven by a vertically invariant horizontal body force. This system admits vertically invariant solutions that satisfy the 2D Navier–Stokes equation. At high Reynolds number and without global rotation, such solutions are usually unstable to three-dimensional perturbations. By contrast, for strong enough global rotation, we prove rigorously that the 2D (and possibly turbulent) solutions are stable to vertically dependent perturbations. We first consider the 3D rotating Navier–Stokes equation linearized around a statistically steady 2D flow solution. We show that this base flow is linearly stable to vertically dependent perturbations when the global rotation is fast enough: under a Reynolds-number-dependent threshold value $Ro_{c}(Re)$ of the Rossby number, the flow becomes exactly 2D in the long-time limit, provided that the initial 3D perturbations are small. We call this property linear two-dimensionalization. We compute explicit lower bounds on $Ro_{c}(Re)$ and therefore determine regions of the parameter space $(Re,Ro)$ where such exact two-dimensionalization takes place. We present similar results in terms of the forcing strength instead of the root-mean-square velocity: the global attractor of the 2D Navier–Stokes equation is linearly stable to vertically dependent perturbations when the forcing-based Rossby number $Ro^{(f)}$ is lower than a Grashof-number-dependent threshold value $Ro_{c}^{(f)}(Gr)$ . We then consider the fully nonlinear 3D rotating Navier–Stokes equation and prove absolute two-dimensionalization: we show that, below some threshold value $Ro_{\mathit{abs}}^{(f)}(Gr)$ of the forcing-based Rossby number, the flow becomes two-dimensional in the long-time limit, regardless of the initial condition (including initial 3D perturbations of arbitrarily large amplitude). These results shed some light on several fundamental questions of rotating turbulence: for arbitrary Reynolds number $Re$ and small enough Rossby number, the system is attracted towards purely 2D flow solutions, which display no energy dissipation anomaly and no cyclone–anticyclone asymmetry. Finally, these results challenge the applicability of wave turbulence theory to describe stationary rotating turbulence in bounded domains.