Sparsifying the resolvent forcing mode via gradient-based optimisation

We consider the use of sparsity-promoting norms in obtaining localised forcing structures from resolvent analysis. By formulating the optimal forcing problem as a Riemannian optimisation, we are able to maximise cost functionals whilst maintaining a unit-energy forcing. Taking the cost functional to be the energy norm of the driven response results in a traditional resolvent analysis and is solvable by a singular value decomposition (SVD). By modifying this cost functional with the L-1-norm, we target spatially localised structures that provide an efficient amplification in the energy of the response. We showcase this optimisation procedure on two flows: plane Poiseuille flow at Reynolds number Re = 4000, and turbulent flow past a NACA 0012 aerofoil at Re = 23 000. In both cases, the optimisation yields sparse forcing modes that maintain important features of the structures arising from an SVD in order to provide a gain in energy. These results showcase the benefits of utilising a sparsity-promoting resolvent formulation to uncover sparse forcings, specifically with a view to using them as actuation locations for flow control.

Automated summary

This paper considers the use of sparsity-promoting norms to obtain localized forcing structures from resolvent analysis. By formulating the optimal forcing problem as a Riemannian optimization, the authors are able to maximize cost functionals while maintaining a unit-energy forcing. They demonstrate this optimization procedure on two different flow cases and show that it produces sparse forcing modes that maintain important features of the structures arising from an SVD. The results highlight the benefits of utilizing a sparsity-promoting resolvent formulation in flow control applications.