Dynamics and thermodynamics of axisymmetric flows: Theory

Abstract

We develop new variational principles to study the structure and the stability of equilibrium states of axisymmetric flows. We show that the axisymmetric Euler equations for inviscid flows admit an infinite number of steady state solutions. We find their general form and provide analytical solutions in some special cases. The system can be trapped in one of these steady states as a result of an inviscid violent relaxation. We show that the stable steady states maximize a (nonuniversal) H-function while conserving energy, helicity, circulation and angular momentum (robust constraints). This can be viewed as a form of generalized selective decay principle. We derive relaxation equations which can be used as numerical algorithm to construct nonlinearly dynamically stable stationary solutions of axisymmetric flows. We also develop a thermodynamical approach to predict the equilibrium state at some fixed coarse-grained scale. We show that the resulting distribution can be divided in two parts: one universal coming from the conservation of robust invariants and one non-universal determined by the initial conditions through the fragile invariants (for freely evolving systems) or by a prior distribution encoding non-ideal effects such as viscosity, small-scale forcing and dissipation (for forced systems). Finally, we derive a parameterization of inviscid mixing to describe the dynamics of the system at the coarse-grained scale. A conceptual interest of this axisymmetric model is to be intermediate between 2D and 3D turbulence.

Fluctuation-Dissipation Relations and Statistical Temperatures in a Turbulent von Kármán Flow

“…Furthermore, r −1 ∂ r r −1 ∂ r ψ + r −2 ∂ 2 z ψ = −ξ. In the inviscid, force-free limit, the global quantities conserved by the Euler equation are the energy E, the generalized helicities H f and the Casimirs I g , given by [7]: …”

“…Advances in that direction have been recently made for flows with symmetries (2D [6], axisymmetric [7]) using tools developed independently by Robert and Sommeria and Miller [8]. They consider freely evolving flows described by the Euler equation (no forcing and no dissipation).…”

“…For Beltrami flows, in which vorticity is proportional to velocity everywhere, i.e. u = λ ∇ × u, the relevant conserved quantities are the energy E and the helicity H so that f (σ) = σ and g = 0 [7]. Let us now apply the statistical mechanics approach introduced in the previous paragraph to a Beltrami flow.…”

“…Let us now apply the statistical mechanics approach introduced in the previous paragraph to a Beltrami flow. The detailed procedure is described in [7,11]. The mixing entropy of the flow S[ρ] is defined using the probability density ρ(σ, ξ, r) to have a certain couple of values for σ and ξ at each position r. Because of the Beltrami hypothesis, the steady state of the flow maximizes S[ρ] at fixed energy E and helicity H. Then, writing δS − βδE − µδH = 0 (where β −1 is a temperature and µ an "helical potential") and using two different mean field approximations, one finds two relations for the averaged fields [14]:…”

Turbulence modeling of the Von Kármán flow: Viscous and inertial stirrings

“…Monchaux et al [22] investigated the properties of the mean and most probable velocity fields in the same configuration. They showed that these two fields are described by two families of functions [23] depending on both the viscosity and the forcing. For large values of the Reynolds number, in some regions, the flow behaves like a Beltrami flow in which vorticity is locally aligned with velocity.…”