Searching for invariant solutions to wall-bounded flows using resolvent-based optimisation
Searching for invariant solutions to wall-bounded flows using resolvent-based optimisation
We present a robust optimisation framework for computing invariant solutions of wall-bounded flows by recasting the Navier–Stokes equations as a variational problem as established in Ashtari & Schneider, JFM (2023). The approach minimises the residual of the governing equations over a finite time horizon, seeking periodic or equilibrium solutions. A novel contribution is made by including a Galerkin projection onto a basis of divergence-free modes that satisfy the no-slip boundary conditions. This projection not only makes the variational framework applicable to wall-bounded flows but it also yields a low-order representation of the dynamics. The basis is derived from resolvent analysis, which provides an orthonormal set. We demonstrate the method on a streamwise invariant formulation of rotating plane Couette flow, obtaining exact equilibrium and periodic solutions consistent with direct numerical simulations. The conditioning of the optimisation problem is analysed in detail, showing that convergence rates depend on the stability properties of the targeted solutions. Finally, we highlight a direct link between the conditioning of the optimisation and the structure of the resolvent operator, suggesting a unifying perspective on both the efficiency of the optimisation and the dynamical significance of resolvent modes.
Burton, Thomas
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Lasagna, Davide
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Symon, Sean
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Burton, Thomas
0d520656-8ff9-4b39-8a06-bae95b8a9faa
Lasagna, Davide
0340a87f-f323-40fb-be9f-6de101486b24
Symon, Sean
2e1580c3-ba27-46e8-9736-531099f3d850
Burton, Thomas, Lasagna, Davide and Symon, Sean
(2026)
Searching for invariant solutions to wall-bounded flows using resolvent-based optimisation.
Journal of Fluid Mechanics.
(In Press)
Abstract
We present a robust optimisation framework for computing invariant solutions of wall-bounded flows by recasting the Navier–Stokes equations as a variational problem as established in Ashtari & Schneider, JFM (2023). The approach minimises the residual of the governing equations over a finite time horizon, seeking periodic or equilibrium solutions. A novel contribution is made by including a Galerkin projection onto a basis of divergence-free modes that satisfy the no-slip boundary conditions. This projection not only makes the variational framework applicable to wall-bounded flows but it also yields a low-order representation of the dynamics. The basis is derived from resolvent analysis, which provides an orthonormal set. We demonstrate the method on a streamwise invariant formulation of rotating plane Couette flow, obtaining exact equilibrium and periodic solutions consistent with direct numerical simulations. The conditioning of the optimisation problem is analysed in detail, showing that convergence rates depend on the stability properties of the targeted solutions. Finally, we highlight a direct link between the conditioning of the optimisation and the structure of the resolvent operator, suggesting a unifying perspective on both the efficiency of the optimisation and the dynamical significance of resolvent modes.
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Submitted date: 6 April 2026
Accepted/In Press date: 18 May 2026
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Local EPrints ID: 511615
URI: http://eprints.soton.ac.uk/id/eprint/511615
ISSN: 0022-1120
PURE UUID: cac9a0f2-9275-4db6-bea2-e47135df3397
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Date deposited: 26 May 2026 16:32
Last modified: 27 May 2026 02:13
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Author:
Thomas Burton
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