In fluid-dynamic stability problems or flow-control studies, the first step is to determine the steady solution of the governing equations. When the steady solution becomes unstable with the variation of a parameter, the numerical method must be able to follow it across the neutral point into the unstable range. When the bifurcation is unsteady, standard relaxation methods exploiting the temporal evolution of the velocity field are unable to do this. On the other hand, since the discretization of the Navier–Stokes equations often leads to a very large discrete system, Newton’s algorithm involving matrix inversion would have too large memory and time requirements to be feasible. An alternative is offered by Newton–Krylov methods, such as, for instance, those based on the GMRES method. We present a new algorithm, in the category of Krylov-subspace methods, to compute unstable steady states. This method, like GMRES, is based on the minimization of the residual norm at each integration step. However, unlike GMRES, the projection basis is updated at each iteration rather than at periodic restarts. The algorithm is sufficiently streamlined to make this possible without negatively impacting on the computation time, moreover it is easily inserted into a pre-existing relaxation procedure as a single black-box subroutine. Results will be shown for the Lorenz problem, the two-dimensional and three-dimensional Navier– Stokes problems. In addition, our method can be used to stabilize periodic orbits and to accelerate the convergence of problems characterized by slowly decaying eigenmodes. In general, BoostConv can be combined with any spatial discretization method, be it a finite-difference, finite-volume, finite-element or spectral method.

BoostConv: finding unstable solutions of the equations of fluid motion

GIANNETTI, Flavio;LUCHINI, Paolo;CITRO, VINCENZO;
2015

Abstract

In fluid-dynamic stability problems or flow-control studies, the first step is to determine the steady solution of the governing equations. When the steady solution becomes unstable with the variation of a parameter, the numerical method must be able to follow it across the neutral point into the unstable range. When the bifurcation is unsteady, standard relaxation methods exploiting the temporal evolution of the velocity field are unable to do this. On the other hand, since the discretization of the Navier–Stokes equations often leads to a very large discrete system, Newton’s algorithm involving matrix inversion would have too large memory and time requirements to be feasible. An alternative is offered by Newton–Krylov methods, such as, for instance, those based on the GMRES method. We present a new algorithm, in the category of Krylov-subspace methods, to compute unstable steady states. This method, like GMRES, is based on the minimization of the residual norm at each integration step. However, unlike GMRES, the projection basis is updated at each iteration rather than at periodic restarts. The algorithm is sufficiently streamlined to make this possible without negatively impacting on the computation time, moreover it is easily inserted into a pre-existing relaxation procedure as a single black-box subroutine. Results will be shown for the Lorenz problem, the two-dimensional and three-dimensional Navier– Stokes problems. In addition, our method can be used to stabilize periodic orbits and to accelerate the convergence of problems characterized by slowly decaying eigenmodes. In general, BoostConv can be combined with any spatial discretization method, be it a finite-difference, finite-volume, finite-element or spectral method.
978-88-97752-55-4
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11386/4668300
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