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首页> 外文期刊>Bulletin of the American Physical Society >APS -70th Annual Meeting of the APS Division of Fluid Dynamics- Event - One-dimensional hydrodynamic equation generating turbulent scaling laws and self-similar singular solutions
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APS -70th Annual Meeting of the APS Division of Fluid Dynamics- Event - One-dimensional hydrodynamic equation generating turbulent scaling laws and self-similar singular solutions

机译：APS-APS流体动力学分部第70届年会-事件-产生湍流比例定律和自相似奇异解的一维流体动力学方程

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摘要

One of the remarkable features of the stochastic laws of fluid turbulence is the emergence of the inertial range in the energy density spectrum on which the energy cascades self-similarly. In the Kolmogorov's theory of fluid turbulence, it is suggested by Onsager that singular solutions of the Navier-Stokes equations in the inviscid limit or the Euler equations that does not conserve the energy play an important role. The existence of energy dissipating weak solution of the Euler equations with $1/3$-H"{o}lder continuity has recently been established by Buckmaster et al. Nevertheless, it remains a theoretical challenge to deduce the stochastic laws from such singular solutions. To gain an insight into this problem, we propose a one-dimensional hydrodynamic nonlinear equation based on the Constantin-Lax-Majda-DeGregorio model. The equation admits an invariant quantity and a finite-time blowup solution in the inviscid case, while with the viscous term and a steady forcing, we obtain a singular steady solution in its inviscid limit. In addition, there emerges the inertial range corresponding to the cascade of the inviscid invariant under a random forcing. In the presentation, we provide recent results on the relation between the turbulent stochastic laws and the singular solutions.
机译：流体湍流随机定律的显着特征之一是能量密度谱中惯性范围的出现，在该惯性范围中能量自相似地级联。在Kolmogorov的流体湍流理论中，Onsager提出，无粘极限中的Navier-Stokes方程的奇异解或不节约能量的Euler方程起着重要的作用。 Buckmaster等人最近建立了具有$ 1/3 $ -H“ {o} lder连续性的Euler方程耗能弱解的存在。然而，从这样的奇异解推导随机定律仍然是一个理论挑战。为了深入了解这个问题，我们在Constantin-Lax-Majda-DeGregorio模型的基础上提出了一维流体动力学非线性方程，该方程在不粘滞的情况下允许不变量和有限时间的爆破解，而对于粘性项和稳态强迫，在其无粘性极限下得到奇异的稳态解，此外，在随机强迫下，出现了与无粘性不变的级联对应的惯性范围，在表述中，我们提供了关于该关系的最新结果在湍流随机定律和奇异解之间。

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