论文标题
Shell模型与3D Navier-Stokes方程有多近？
How close are shell models to the 3D Navier-Stokes equations?
论文作者
论文摘要
壳模型已经在流体动力湍流的研究中发现了广泛的应用，因为它们即使在很大的雷诺数上也很容易在数值上求解。尽管存在空间变化，但它们还是准确地重现了完全发达的均质和各向同性湍流的主要统计特性。此外，它们享有规律性的属性，这些属性仍然为三维（3D）Navier-Stokes方程（NSES）保持开放。这项研究的目的是对Shell模型和NSE进行严格的比较。事实证明，在两个系统中，平均能量耗散率的估计值相同。与3D NSE相比，对壳模型的速度及其高阶衍生物的估计值显示出较弱的雷诺数依赖性。实际上，发现壳模型的速度衍生估计值相当于与3D Navier-Stokes方程（VGA-NSES）的速度梯度平均版本相对应的速度估计值，而速度估计值甚至更轻。在雷诺数广泛的范围内的数值模拟证实了壳模型的估计值。
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.
