A geometric interpretation of coherent structures in Navier–Stokes flows

Roulstone, Ian; Banos, Bertrand; Gibbon, John; Roubtsov, Vladimir

doi:10.1098/rspa.2008.0483

Cited by 8 publications

(29 citation statements)

References 26 publications

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“…We now proceed to show how a geometry, very similar to the one described earlier, can be recovered for three-dimensional flows with symmetry. Our results extend those of Roulstone et al (2009b).…”

Section: The Symplectic Reductionsupporting

confidence: 90%

Monge–Ampère structures and the geometry of incompressible flows

Banos¹,

Roubtsov²,

Roulstone³

2016

J. Phys. A: Math. Theor.

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We show how a symmetry reduction of the equations for incompressible hydrodynamics in three dimensions leads naturally to MongeAmpère structure, and Burgers'-type vortices are a canonical class of solutions associated with this structure. The mapping of such solutions, which are characterised by a linear dependence of the third component of the velocity on the coordinate defining the axis of rotation, to solutions of the incompressible equations in two dimensions is also shown to be an example of a symmetry reduction. The MongeAmpère structure for incompressible flow in two dimensions is shown to be hyper-symplectic.

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Section: The Symplectic Reductionsupporting

confidence: 90%

Monge–Ampère structures and the geometry of incompressible flows

Banos¹,

Roubtsov²,

Roulstone³

2016

J. Phys. A: Math. Theor.

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“…The appearance of a Monge-Ampère equation for two-dimensional incompressible flows led [7] to study this problem from the point of view of the Monge-Ampère geometry of [8].…”

Section: Introductionmentioning

confidence: 99%

“…Focusing on two-dimensional incompressible flows, in Section 2.2 we introduce the machinery of Monge-Ampère geometry and Monge-Ampère structures, following the geometric approach as described, for example, by [8]. This allows us to formulate the Monge-Ampère equation arising in the Poisson equation for the pressure, revisiting some of the results of [7].…”

Section: Introductionmentioning

confidence: 99%

Monge-Ampere Geometry and Vortices

Napper¹,

Roulstone²,

Rubtsov³

et al. 2023

Preprint

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We introduce a new approach to Monge-Ampère geometry based on techniques from higher symplectic geometry. Our work is motivated by the application of Monge-Ampère geometry to the Poisson equation for the pressure that arises for incompressible Navier-Stokes flows. Whilst this equation constitutes an elliptic problem for the pressure, it can also be viewed as a non-linear partial differential equation connecting the pressure, the vorticity, and the rate-of-strain. As such, it is a key diagnostic relation in the quest to understand the formation of vortices in turbulent flows. We study this equation via the (higher) Lagrangian submanifold it defines in the cotangent bundle to the configuration space of the fluid. Using our definition of a (higher) Monge-Ampère structure, we study an associated metric on the cotangent bundle together with its pull-back to the (higher) Lagrangian submanifold. The signatures of these metrics are dictated by the relationship between vorticity and rate-of-strain, and their scalar curvatures can be interpreted in a physical context in terms of the accumulation of vorticity, strain, and their gradients. We show explicity, in the case of two-dimensional flows, how topological information can be derived from the Monge-Ampère geometry of the Lagrangian submanifold. We also demonstrate how certain solutions to the three-dimensional incompressible Navier-Stokes equations, such as Hill's spherical vortex and an integrable case of Arnol'd-Beltrami-Childress flow, have symmetries that facilitate a formulation of these solutions from the perspective of (higher) symplectic reduction.

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“…Although it would be premature to attempt to give an unequivocal answer, Roulstone et al (2009b) have shown how the almost-complex structures can be used to describe coherent structures in Navier-Stokes flows in three dimensions. The salient point of this work is that the complex geometry provides a framework for studying coherent structures that is not readily accessible via the more traditional analysis of the underlying partial differential equations that can be performed for incompressible flows in two dimensions (e.g.…”

Section: Discussionmentioning

confidence: 99%