Variational methods for finding periodic orbits in the incompressible Navier–Stokes equations

Parker, Jeremy; Schneider, Tobias

doi:10.1017/jfm.2022.299

Cited by 8 publications

(15 citation statements)

References 27 publications

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“…First, we apply the adjoint-based variational method to 3-D wall-bounded flows. Previously, the variational approach had been successfully applied only to a 2-D Kolmogorov flow in a doubly periodic domain without walls (Farazmand 2016; Parker & Schneider 2022). The primary challenge in extending the variational method for computing equilibria to wall-bounded flows lies in handling the nonlinear, non-local pressure in the presence of solid walls.…”

Section: Summary and Concluding Remarksmentioning

confidence: 99%

“…The advantages of the adjoint-based variational method have inspired its application in computing other invariant sets, such as periodic orbits (Azimi, Ashtari & Schneider 2022; Parker & Schneider 2022) and connecting orbits (Ashtari & Schneider 2023). These methods view the identification of a periodic or connecting orbit as a minimisation problem in the space of space–time fields with prescribed behaviour in the temporal direction.…”

Section: Summary and Concluding Remarksmentioning

confidence: 99%

“…This improvement, however, is obtained by sacrificing the quadratic convergence of the Newton iterations and exhibiting slow convergence. In the context of fluid mechanics, the variational approach has been applied successfully to the 2-D Kolmogorov flows (see Farazmand 2016; Parker & Schneider 2022).…”

Section: Introductionmentioning

confidence: 99%

“…Beyond the high-dimensionality of the 3-D wall-bounded flows, the main challenge in the application of this method lies in handling the wall boundary conditions that cannot be imposed readily while evolving the adjoint-descent dynamics (see § 3.3). This is in contrast to doubly periodic 2-D (or triply periodic 3-D) flows where the adjoint-descent dynamics is subject to periodic boundary conditions only, that can be imposed by representing variables in Fourier basis (Farazmand 2016; Parker & Schneider 2022). To construct a suitable formulation for 3-D flows in the presence of walls, we project the evolving velocity field onto the space of divergence-free fields and constrain pressure so that it satisfies the pressure Poisson equation instead of evolving independent of the velocity (see § 3.4).…”

Section: Introductionmentioning

confidence: 99%

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Identifying invariant solutions of wall-bounded three-dimensional shear flows using robust adjoint-based variational techniques

Ashtari,

Schneider

2023

J. Fluid Mech.

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Invariant solutions of the Navier–Stokes equations play an important role in the spatiotemporally chaotic dynamics of turbulent shear flows. Despite the significance of these solutions, their identification remains a computational challenge, rendering many solutions inaccessible and thus hindering progress towards a dynamical description of turbulence in terms of invariant solutions. We compute equilibria of three-dimensional wall-bounded shear flows using an adjoint-based matrix-free variational approach. To address the challenge of computing pressure in the presence of solid walls, we develop a formulation that circumvents the explicit construction of pressure and instead employs the influence matrix method. Together with a data-driven convergence acceleration technique based on dynamic mode decomposition, this yields a practically feasible alternative to state-of-the-art Newton methods for converging equilibrium solutions. We compute multiple equilibria of plane Couette flow starting from inaccurate guesses extracted from a turbulent time series. The variational method outperforms Newton(-hookstep) iterations in converging successfully from poor initial guesses, suggesting a larger convergence radius.

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Section: Summary and Concluding Remarksmentioning

confidence: 99%

Section: Summary and Concluding Remarksmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Identifying invariant solutions of wall-bounded three-dimensional shear flows using robust adjoint-based variational techniques

Ashtari,

Schneider

2023

J. Fluid Mech.

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“…and minimizing this error term by any suitable variational or optimization method, possibly in conjunction with a high-dimensional variant of the Newton method [54,89,113,144,178].…”

Section: Computing Lattice States For Nonlinear Theoriesmentioning

confidence: 99%

A chaotic lattice field theory in one dimension*

Liang¹,

Cvitanović²

2022

J. Phys. A: Math. Theor.

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Motivated by Gutzwiller's semiclassical quantization, in which unstable periodic orbits of low-dimensional deterministic dynamics serve as a WKB `skeleton' for chaotic quantum mechanics, we construct the corresponding deterministic skeleton for infinite-dimensional lattice-discretized scalar field theories. In the field-theoretical formulation, there is no evolution in time, and there is no `Lyapunov horizon'; there is only an enumeration of lattice states that contribute to the theory's partition sum, each a global spatiotemporal solution of system's deterministic Euler-Lagrange equations. The reformulation aligns `chaos theory' with the standard solid state, field theory, and statistical mechanics. In a spatiotemporal, crystallographer formulation, the time-periodic orbits of dynamical systems theory are replaced by periodic d-dimensional Bravais cell tilings of spacetime, each weighted by the inverse of its instability, its Hill determinant. Hyperbolic shadowing of large cells by smaller ones ensures that the predictions of the theory are dominated by the smallest Bravais cells. The form of the partition function of a given field theory is determined by the group of its spatiotemporal symmetries, that is, by the space group of its lattice discretization, best studied on its reciprocal lattice. Already 1-dimensional lattice discretization is of sufficient interest to be the focus of this paper. In particular, from a spatiotemporal field theory perspective, `time'-reversal is a purely crystallographic notion, a reflection point group, leading to a novel, symmetry quotienting perspective of time-reversible theories and associated topological zeta functions.

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Multi-scale invariant solutions in plane Couette flow: a reduced-order model approach

McCormack,

Cavalieri,

Hwang

2024

J. Fluid Mech.

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Plane Couette flow at Reynolds number $Re=1200$ (based on the channel half-height and half the velocity difference between the top and bottom plates) is investigated with a spatial domain designed to retain only two spanwise integral length scales. In this system, the computation of invariant solutions that are physically representative of the turbulent state has been understood to be challenging. To address this challenge, our approach is to employ an accurate reduced-order model with 600 degrees of freedom (Cavalieri & Nogueira, Phys. Rev. Fluids, vol. 7, 2022, L102601). Using the two-scale energy budget and the temporal cross-correlation of key observables, it is first demonstrated that the model contains most of the multi-scale physical processes identified recently (Doohan et al., J. Fluid Mech., vol. 913, 2021, A8); i.e. the large- and small-scale self-sustaining processes, the energy cascade for turbulent dissipation, and an energy-cascade mediated small-scale production mechanism. Invariant solutions of the reduced-order model are subsequently computed, including 96 equilibria and 43 periodic orbits. It is found that none of the computed equilibrium solutions are able to reproduce an accurate energy balance associated with the multi-scale dynamics of the turbulent state. Incorporation of unsteadiness into invariant solutions is seen to be essential for a sensible description of the multi-scale turbulent dynamics and the related energetics, at least in this type of flow, as periodic orbits with a sufficiently long period are mainly able to describe the complex spatio-temporal dynamics associated with the known multi-scale phenomena.

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Variational methods for finding periodic orbits in the incompressible Navier–Stokes equations

Cited by 8 publications

References 27 publications

Identifying invariant solutions of wall-bounded three-dimensional shear flows using robust adjoint-based variational techniques

Identifying invariant solutions of wall-bounded three-dimensional shear flows using robust adjoint-based variational techniques

A chaotic lattice field theory in one dimension*

Multi-scale invariant solutions in plane Couette flow: a reduced-order model approach

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