scholarly journals Long-time states of inverse cascades in the presence of a maximum length scale

1983 ◽  
Vol 30 (3) ◽  
pp. 479-493 ◽  
Author(s):  
Murshed Hossain ◽  
William H. Matthaeus ◽  
David Montgomery
Keyword(s):  
Dissipation Rate ◽  
Maximum Length ◽  
Navier Stokes ◽  
Asymptotic State ◽  
Length Scale ◽  
Two Dimensional ◽  
Inverse Cascade ◽  
Long Time

It is shown numerically, both for the two-dimensional Navier-Stokes (guidingcentre plasma) equations and for two-dimensional magnetohydrodynamics, that the long-time asymptotic state in a forced inverse-cascade situation is one in which the spectrum is completely dominated by its own fundamental. The growth continues until the fundamental is dissipatively limited by its own dissipation rate.

scholarly journals Kármán–Howarth theorem for the Lagrangian-averaged Navier–Stokes–alpha model of turbulence

2002 ◽  
Vol 467 ◽  
pp. 205-214 ◽  
Author(s):  
DARRYL D. HOLM
Keyword(s):  
Energy Spectrum ◽  
Dissipation Rate ◽  
Isotropic Turbulence ◽  
Energy Dissipation Rate ◽  
Navier Stokes ◽  
Length Scale ◽  
Two Dimensional ◽  
Homogeneous Isotropic Turbulence ◽  
Third Order ◽  
Kinetic Energy Spectrum

The Lagrangian averaged Navier–Stokes–alpha (LANS-α) model of turbulence is found to possess a Kármán–Howarth (KH) theorem for the dynamics of its second-order autocorrelation functions in homogeneous isotropic turbulence. This KH result implies that alpha-filtering in the LANS-α model of turbulence does not affect the exact Navier–Stokes relation between second and third moments at separation distances large compared to the model's length scale α. Moreover, at separations r that are smaller than α, the KH scaling between energy dissipation rate and longitudinal third-order autocorrelation changes to match the scaling found in two-dimensional incompressible flow. This is consistent with the corresponding change in scaling of the kinetic energy spectrum from k−5/3 for larger scales with kα < 1, which switches to k−3 for smaller scales with kα > 1, as discovered in Foias, Holm & Titi (2001).

Three-dimensional impulsive vortex formation from slender orifices

2011 ◽  
Vol 666 ◽  
pp. 506-520 ◽  
Author(s):  
F. DOMENICHINI
Keyword(s):  
Stokes Equations ◽  
Intermediate Phase ◽  
Vortex Formation ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Vortex Ring Formation ◽  
Long Time ◽  
Definition Of

The vortex formation behind an orifice is a widely investigated phenomenon, which has been recently studied in several problems of biological relevance. In the case of a circular opening, several works in the literature have shown the existence of a limiting process for vortex ring formation that leads to the concept of critical formation time. In the different geometric arrangement of a planar flow, which corresponds to an opening with straight edges, it has been recently outlined that such a concept does not apply. This discrepancy opens the question about the presence of limiting conditions when apertures with irregular shape are considered. In this paper, the three-dimensional vortex formation due to the impulsively started flow through slender openings is studied with the numerical solution of the Navier–Stokes equations, at values of the Reynolds number that allow the comparison with previous two-dimensional findings. The analysis of the three-dimensional results reveals the two-dimensional nature of the early vortex formation phase. During an intermediate phase, the flow evolution appears to be driven by the local curvature of the orifice edge, and the time scale of the phenomena exhibits a surprisingly good agreement with those found in axisymmetric problems with the same curvature. The long-time evolution shows the complete development of the three-dimensional vorticity dynamics, which does not allow the definition of further unifying concepts.

scholarly journals Metastability and rapid convergence to quasi-stationary bar states for the two-dimensional Navier–Stokes equations

2013 ◽  
Vol 143 (5) ◽  
pp. 905-927 ◽  
Author(s):  
Margaret Beck ◽  
C. Eugene Wayne
Keyword(s):  
Decay Rate ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Optimal Decay ◽  
Optimal Decay Rate ◽  
Long Time ◽  
High Reynolds

Quasi-stationary, or metastable, states play an important role in two-dimensional turbulent fluid flows, where they often emerge on timescales much shorter than the viscous timescale, and then dominate the dynamics for very long time intervals. In this paper we propose a dynamical systems explanation of the metastability of an explicit family of solutions, referred to as bar states, of the two-dimensional incompressible Navier–Stokes equation on the torus. These states are physically relevant because they are associated with certain maximum entropy solutions of the Euler equations, and they have been observed as one type of metastable state in numerical studies of two-dimensional turbulence. For small viscosity (high Reynolds number), these states are quasi-stationary in the sense that they decay on the slow, viscous timescale. Linearization about these states leads to a time-dependent operator. We show that if we approximate this operator by dropping a higher-order, non-local term, it produces a decay rate much faster than the viscous decay rate. We also provide numerical evidence that the same result holds for the full linear operator, and that our theoretical results give the optimal decay rate in this setting.

scholarly journals Well-Posedness and Long-Time Behavior of Solutions for Two-Dimensional Navier-Stokes Equations with Infinite Delay and General Hereditary Memory

2021 ◽  
Author(s):  
Yasi Zheng ◽  
Wenjun Liu ◽  
Yadong Liu
Keyword(s):  
Stokes Equations ◽  
Infinite Delay ◽  
Navier Stokes ◽  
Time Behavior ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Bounded Absorbing Set ◽  
Long Time ◽  
The One ◽  
Class Of Functions

We address the dynamics of two-dimensional Navier-Stokes models with infinite delay and hereditary memory, whose kernels are a much larger class of functions than the one considered in the literature, on a bounded domain. We prove the existence and uniqueness of weak solutions by means of Faedo-Galerkin method. Moreover, we establish the existence of global attractor for the system with the existence of a bounded absorbing set and asymptotic compact property.

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