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.
D.
Barkley
, “
Linear analysis of the cylinder wake mean flow
,”
Europhys. Lett.
75
(
12
),
750
(
2006
).
2.
E.
Åkervik
,
L.
Brandt
,
D. S.
Henningson
,
J.
Hœpffner
,
O.
Marxen
, and
P.
Schlatter
, “
Steady solutions of the Navier-Stokes equations by selective frequency damping
,”
Phys. Fluids
18
,
068102
(
2006
).
3.
B.
Pier
, “
Local and global instabilities in the wake of a sphere
,”
J. Fluid Mech.
603
,
39
61
(
2008
).
4.
S.
Bagheri
,
P.
Schlatter
,
P. J.
Schmid
, and
D. S.
Henningson
, “
Global instability of a jet in crossflow
,”
J. Fluid Mech.
624
,
33
44
(
2009
).
5.
E.
Åkervik
,
J.
Hoepffner
,
U.
Ehrenstein
, and
D. S.
Henningson
, “
Optimal growth, model reduction and control in a separated boundary-layer flow using global eigenmodes
,”
J. Fluid Mech.
579
(
1
),
305
314
(
2007
).
6.
L. E.
Jones
and
R. D.
Sandberg
, “
Numerical analysis of tonal airfoil self-noise and acoustic feedback-loops
,”
J. Sound Vib.
330
(
25
),
6137
6152
(
2011
).
7.
E.
Vyazmina
, “
Bifurcations in a swirling flow
,” Ph.D. thesis (
École Polytechnique
,
2010
).
8.
J.
Kim
and
T. R.
Bewley
, “
A linear systems approach to flow control
,”
Annu. Rev. Fluid Mech.
39
,
383
417
(
2007
).
9.
C. D.
Pruett
,
T. B.
Gatski
,
C. E.
Grosch
, and
W. D.
Thacker
, “
The temporally filtered Navier-Stokes equations: Properties of the residual stress
,”
Phys. Fluids
15
(
8
),
2127
(
2003
).
10.
C. D.
Pruett
,
B. C.
Thomas
,
C. E.
Grosch
, and
T. B.
Gatski
, “
A temporal approximate deconvolution model for large-eddy simulation
,”
Phys. Fluids
18
,
028104
(
2006
).
11.
L. S.
Tuckerman
and
D.
Barkley
,
Bifurcation Analysis for Timesteppers
(
Springer
,
2000
).
12.
I.
Faragó
, “
Splitting methods and their application to the abstract Cauchy problems
,”
Numerical Analysis and Its Applications
(
Springer
,
2005
), pp.
35
45
.
13.
C.
Norberg
, “
Fluctuating lift on a circular cylinder: Review and new measurements
,”
J. Fluids Struct.
17
,
57
96
(
2003
).
14.
See http://www.nektar.info for Nektar++ (last accessed 2014).
15.
G. E.
Karniadakis
and
S. J.
Sherwin
,
Spectral/hp Element Methods for CFD
, 2nd ed. (
Oxford University Press
,
2005
).
16.
U. M.
Ascher
,
S. J.
Ruuth
, and
B. T. R.
Wetton
, “
Implicit-explicit methods for time-dependent partial differential equations
,”
SIAM J. Numer. Anal.
32
(
3
),
797
823
(
1995
).
17.
S. J.
Sherwin
and
H. M.
Blackburn
, “
Three-dimensional instabilities and transition of steady and pulsatile axisymmetric stenotic flows
,”
J. Fluid Mech.
533
,
297
327
(
2005
).
You do not currently have access to this content.