Encapsulated formulation of the selective frequency damping method

We present an alternative “encapsulated” formulation of the selective frequency damping method for finding unstable equilibria of dynamical systems, which is particularly useful when analysing the stability of fluid flows. The formulation makes use of splitting methods, which means that it can be wrapped around an existing time-stepping code as a “black box.” The method is first applied to a scalar problem in order to analyse its stability and highlight the roles of the control coefficient χ and the filter width Δ in the convergence (or not) towards the steady-state. Then the steady-state of the incompressible flow past a two-dimensional cylinder at Re = 100, obtained with a code which implements the spectral/hp element method, is presented.

[1]  Dan S. Henningson,et al.  Global stability of a jet in crossflow , 2009, Journal of Fluid Mechanics.

[2]  L. Tuckerman,et al.  Bifurcation Analysis for Timesteppers , 2000 .

[3]  O. Marxen,et al.  Steady solutions of the Navier-Stokes equations by selective frequency damping , 2006 .

[4]  Thomas B. Gatski,et al.  The temporally filtered Navier–Stokes equations: Properties of the residual stress , 2003 .

[5]  Spencer J. Sherwin,et al.  Three-dimensional instabilities and transition of steady and pulsatile axisymmetric stenotic flows , 2005, Journal of Fluid Mechanics.

[6]  Thomas Bewley,et al.  A Linear Systems Approach to Flow Control , 2007 .

[7]  D. Barkley.  Linear analysis of the cylinder wake mean flow , 2006 .

[8]  István Faragó,et al.  Splitting Methods and Their Application to the Abstract Cauchy Problems , 2004, NAA.

[9]  Thomas B. Gatski,et al.  A Temporal Approximate Deconvolution Model for Large-Eddy Simulation , 2006 .

[10]  Steven J. Ruuth,et al.  Implicit-explicit methods for time-dependent partial differential equations , 1995 .

[11]  C. Norberg.  Fluctuating lift on a circular cylinder: review and new measurements , 2003 .

[12]  T. Schneider,et al.  A Linear Systems Approach to Flow Control , 2007 .

[13]  G. Karniadakis,et al.  Spectral/hp Element Methods for CFD , 1999 .

[14]  Benoît Pier,et al.  Local and global instabilities in the wake of a sphere , 2008, Journal of Fluid Mechanics.

[15]  E. Vyazmina.  Bifurcations in a swirling flow , 2010 .

[16]  D. Henningson,et al.  Optimal growth, model reduction and control in a separated boundary-layer flow using global eigenmodes , 2007, Journal of Fluid Mechanics.

[17]  Richard D. Sandberg,et al.  Numerical analysis of tonal airfoil self-noise and acoustic feedback-loops , 2011 .