How close are shell models to the 3D Navier-Stokes equations?
arXiv:2009.01583 · doi:10.1088/1361-6544/abe096
Abstract
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.
arXiv comment22 pages, 3 figures
References in corpus (16)
- An optimal shell model of turbulence
- Asymptotic Exponents from Low-Reynolds-Number Flows
- Analytic Study of Shell Models of Turbulence
- Anomalous scaling and universality in hydrodynamic systems with power-law forcing
- Inverse Cascade Regime in Shell Models of 2-Dimensional Turbulence
- Vorticity moments in four numerical simulations of the 3D Navier-Stokes equations
- On the Global Regularity of a Helical-decimated Version of the 3D Navier-Stokes Equations
- Weak and strong solutions of the Navier-Stokes equations and their relation to a chessboard of convergent inverse length scales
- Nelkin scaling for the Burgers equation and the role of high-precision calculations
- A dyadic model on a tree
- Ultrametric structure of multiscale energy correlations in turbulent models
- Intermittency, cascades and thin sets in three-dimensional Navier-Stokes turbulence
- Depletion of Nonlinearity in Magnetohydrodynamic Turbulence: Insights from Analysis and Simulations
- The role of BKM-type theorems in Euler, Navier-Stokes and Cahn-Hilliard-Navier-Stokes analysis
- Energy dissipation statistics in a shell model of turbulence
- Sharp Lower Bounds for the Dimension of the Global Attractor of the Sabra Shell Model of Turbulence