壳模型与三维纳维-斯托克斯方程有多接近？
How close are shell models to the 3D Navier-Stokes equations?
摘要：
壳模型在流体动力学湍流研究中得到了广泛的应用，因为它们即使在很大的雷诺数下也很容易数值求解。虽然缺乏空间变化，他们准确地再现充分发展的均匀和各向同性湍流的主要统计特性。此外，他们享有的规则性仍然是开放的三维（3D）纳维-斯托克斯方程（NSE）。本研究的目的是在壳模型和NSE之间进行严格的比较。事实证明，两个系统中只有平均能量耗散率的估计是相同的。速度及其高阶导数的估计显示了较弱的雷诺数的依赖性壳模型比三维NSE。事实上，壳模型的速度导数估计被发现是等价于那些对应于3D纳维-斯托克斯方程（VGA-NSE）的速度梯度平均版本，而速度估计甚至更温和。在很宽的雷诺数范围内的数值模拟证实了壳模型的估计。
英文摘要：
Shell models have found wide application in the study of hydrodynamic turbulence because they are easily solved numerically even at very large Reynolds numbers. Although bereft of spatial variation, they accurately reproduce the main statistical properties of fully-developed homogeneous and isotropic turbulence. Moreover, they enjoy regularity properties which still remain open for the three-dimensional (3D) Navier-Stokes equations (NSEs). The goal of this study is to make a rigorous comparison between shell models and the NSEs. It turns out that only the estimate of the mean energy dissipation rate is the same in both systems. The estimates of the velocity and its higher-order derivatives display a weaker Reynolds number dependence for shell models than for the 3D NSEs. Indeed, the velocity-derivative estimates for shell models are found to be equivalent to those corresponding to a velocity gradient averaged version of the 3D Navier-Stokes equations (VGA-NSEs), while the velocity estimates are even milder. Numerical simulations over a wide range of Reynolds numbers confirm the estimates for shell models.
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