Resolvent-Based Optimisation for Invariant Flows

TL;DR

This work advances the search for invariant solutions in wall-bounded flows by embedding a Galerkin projection onto a divergence-free, no-slip resolvent-basis within a variational optimisation framework. By recasting Navier–Stokes dynamics as a global residual minimisation and projecting onto a resolvent-based basis, the method enforces boundary conditions naturally and yields a low-dimensional representation that still discovers exact equilibria and periodic solutions in a 2D3C RPCF setting. A key contribution is linking the Hessian conditioning of the optimisation to the resolvent spectrum: truncating modes with small singular values preconditions the problem, accelerating convergence while preserving essential flow features. The results suggest this framework can serve as a robust standalone solver and as a preconditioner for Newton-based approaches, with potential for extension to larger, fully 3D wall-bounded flows and parallel implementations for scalability.

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 and 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 2D3C 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.

Figures (15)

• Figure 1: Schematic of an arbitrary loop in state-space that does not satisfy the governing equations as its tangent vector $\partial\bm{u}/\partial t$ is not aligned with the evolution operator $\mathcal{N}=-\left(\bm{u}\cdot\bm{\nabla}\right)\bm{u}+\frac{1}{\mathit{Re}}\bm{\Delta}\bm{u}$.
• Figure 2: Schematic for a Galerkin projection of state-space loop representing a velocity field onto the linear subspace in \ref{['eq:opt-space']}.
• Figure 3: Flow diagram of the computations performed at each iteration of the optimisation.
• Figure 4: Bifurcation diagram of RPCF over a range of Reynolds number and $\mathit{Ro}=0.5$ showing the transition from the stable laminar solution to turbulent flow. Also plotted are the corresponding kinetic energy extrema of the periodic solution obtained from optimisation at $\mathit{Re}=400$ in section \ref{['sec:periodic']}, denoted with red triangles.
• Figure 5: Equilibrium solutions labelled in figure \ref{['fig:dns-bifurcation']} at $\mathit{Re}=50$.