Quasi-stationary states in a circular geometry

Keetels, GH Geert; Clercx, Hjh Herman; Heijst, van Gjf Gert-Jan

doi:10.1016/j.physd.2009.03.011

Cited by 5 publications

(7 citation statements)

References 32 publications

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“…With a nonzero initial angular momentum (|L 0 | > 0) in the flow, the final state usually consists of a single domain-filling cell. This behaviour agrees with observations by Maassen et al (1999Maassen et al ( , 2002 in experiments on decaying stratified turbulence in a circular container and confirmed in numerical studies by the same authors and by Keetels et al (2009). The flow in these experiments was initialized by moving a rake of vertical bars through the stratified fluid along a straight diametrical path.…”

Section: The Effect Of the Domain Shapesupporting

confidence: 90%

“…In a numerical study of decaying 2D turbulence on a circular domain, it was shown by Li and Montgomery (1996) and Li et al (1997) that the boundaries lead to a different decay scenario, somewhat depending on the imposed boundary condition (stress-free or no-slip). Additional support for their observations is found in recent high initial Reynolds number simulations, see Schneider and Farge (2005) and Keetels et al (2008Keetels et al ( , 2009. A similar effect of solid lateral boundaries was observed in the numerical study by Clercx et al (1999) of decaying 2D turbulence on a square domain, with either stress-free or no-slip conditions.…”

Section: The Effect Of Lateral Boundariessupporting

confidence: 78%

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Studies on quasi-2D turbulence—the effect of boundaries

Heijst

Clercx

2009

Fluid Dyn. Res.

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This paper addresses the effects of domain boundaries on the behaviour of quasi-two-dimensional flows, thereby distinguishing between lateral boundaries of horizontal flow domains and the horizontal boundaries confining shallow fluid layers. As already discussed in some recent papers, the lateral walls may play an essential role in acting as sources of filamentary highamplitude vorticity, which usually affects the flow evolution in the interior of the domain. Besides, walls exert forces that may promote the self-organization of the flow, and hence contribute to a change in the net angular momentum of the flow. Both aspects will be reviewed briefly.In contrast to what is commonly assumed, shallow-layer flows may develop an essentially three-dimensional structure, with vertical gradients in the principal horizontal flow field and with significant vertical velocity components. These features have been found recently in experiments on electromagnetically generated flows in a shallow fluid layer. In this paper, we will discuss some of these experimental observations as well as some numerical simulation results that are helpful in explaining the observed flow dynamics.

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Section: The Effect Of the Domain Shapesupporting

confidence: 90%

Section: The Effect Of Lateral Boundariessupporting

confidence: 78%

Studies on quasi-2D turbulence—the effect of boundaries

Heijst

Clercx

2009

Fluid Dyn. Res.

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“…(D21) and using the constraints (D23)-(D24), we easily obtain Ṡ = X 2 D dr, (D32) so that Ṡ ≥ 0 provided that D is positive. On the other hand Ṡ = 0 iff X = 0 leading to the condition of equilibrium (42). From Lyapunov's direct method, we conclude that these relaxation equations tend to a maximum of entropy at fixed energy, circulation and microscopic enstrophy.…”

mentioning

confidence: 81%

“…Our study is more direct and can offer a complementary discussion of the problem. The maximization problem (70) has also been studied recently by Keetels et al [42] with different boundary conditions adapted to viscous flows.…”

Section: E Connection Between Different Variational Principlesmentioning

confidence: 99%

“…In particular, they showed that the point (Γ = 0, τ c = 1.12) corresponds to a bicritical point separating a microcanonical first order transition line to two second order transition lines. Keetels, Clercx & van Heijst (2009) [42] have studied minimum enstrophy states in rectangular or circular domains with boundary conditions taking into account the effect of viscosity. Finally, Taylor, Borchardt & Helander (2009) [43] have shown that the two types of solutions appearing in the study of Chavanis & Sommeria [26] could explain the process of "spin-up" discovered by Clercx, Maassen & van Heijst (1998) [44].…”

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confidence: 99%

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Statistical mechanics of two-dimensional Euler flows and minimum enstrophy states

Naso¹,

Chavanis²,

Dubrulle³

2010

Eur. Phys. J. B

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Dated: To be included later)A simplified thermodynamic approach of the incompressible 2D Euler equation is considered based on the conservation of energy, circulation and microscopic enstrophy. Statistical equilibrium states are obtained by maximizing the Miller-Robert-Sommeria (MRS) entropy under these sole constraints. We assume that these constraints are selected by properties of forcing and dissipation. We find that the vorticity fluctuations are Gaussian while the mean flow is characterized by a linear ω − ψ relationship. Furthermore, we prove that the maximization of entropy at fixed energy, circulation and microscopic enstrophy is equivalent to the minimization of macroscopic enstrophy at fixed energy and circulation. This provides a justification of the minimum enstrophy principle from statistical mechanics when only the microscopic enstrophy is conserved among the infinite class of Casimir constraints. Relaxation equations towards the statistical equilibrium state are derived. These equations can serve as numerical algorithms to determine maximum entropy or minimum enstrophy states. We use these relaxation equations to study geometry induced phase transitions in rectangular domains. In particular, we illustrate with the relaxation equations the transition between monopoles and dipoles predicted by Chavanis & Sommeria [J. Fluid. Mech. 314, 267 (1996)]. We take into account stable as well as metastable states and show that metastable states are robust and have negative specific heats. This is the first evidence of negative specific heats in that context. We also argue that saddle points of entropy can be long-lived and play a role in the dynamics because the system may not spontaneously generate the perturbations that destabilize them. PACS numbers: 05.20.-y Classical statistical mechanics -05.45.-a Nonlinear dynamics and chaos -05.90.+m Other topics in statistical physics, thermodynamics, and nonlinear dynamical systems -47.10.-g General theory in fluid dynamics -47.15.ki Inviscid flows with vorticity -47.20.-k Flow instabilities -47.32.-y Vortex dynamics; rotating fluids

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