This article presents techniques for the analysis of fluid systems. It adopts an optimization-based point of view, formulating common concepts such as stability and receptivity in terms of a cost functional to be optimized subject to constraints given by the governing equations. This approach differs significantly from eigenvalue-based methods that cover the time-asymptotic limit for stability problems or the resonant limit for receptivity problems. Formal substitution of the solution operator for linear time-invariant systems results in the matrix exponential norm and the resolvent norm as measures to assess the optimal response to initial conditions or external harmonic forcing. The optimization-based approach can be extended by introducing adjoint variables that enforce governing equations and constraints. This step allows the analysis of far more general fluid systems, such as time-varying and nonlinear flows, and the investigation of wavemaker regions, structural sensitivities, and passive control strategies.

References

1.
Schmid
,
P. J.
, and
Henningson
,
D. S.
,
2001
,
Stability and Transition in Shear Flows
,
Springer
,
New York
.
2.
Schmid
,
P. J.
,
2007
, “
Nonmodal Stability Theory
,”
Ann. Rev. Fluid Mech.
,
39
, pp.
129
162
.10.1146/annurev.fluid.38.050304.092139
3.
Horn
,
R. A.
, and
Johnson
,
C. R.
,
1994
,
Topics in Matrix Analysis
,
Cambridge University Press
,
Cambridge, UK
.
4.
Trefethen
,
L. N.
,
Trefethen
,
A. N.
,
Reddy
,
S. C.
, and
Driscoll
,
T. A.
,
1993
, “
Hydrodynamic Stability Without Eigenvalues
,”
Science
,
261
, pp.
578
584
.10.1126/science.261.5121.578
5.
Jovanovic
,
M. R.
, and
Bamieh
,
B.
,
2005
, “
Componentwise Energy Amplification in Channel Flows
,”
J. Fluid Mech.
,
534
, pp.
145
184
.10.1017/S0022112005004295
6.
Klinkenberg
,
J.
,
deLange
,
H. C.
, and
Brandt
,
L.
,
2011
, “
Modal and Nonmodal Stability of Particle-Laden Channel Flow
,”
Phys. Fluids
,
23
, p.
064110
.10.1063/1.3599696
7.
Saffman
,
P. G.
,
1962
, “
On the Stability of Laminar Flow of a Dusty Gas
,”
J. Fluid Mech.
,
13
, pp.
120
128
.10.1017/S0022112062000555
8.
Trefethen
,
L. N.
, and
Embree
,
M.
,
2005
,
Spectra and Pseudospectra
,
Princeton University Press
,
Princeton, NJ
.
9.
Andersson
,
P.
,
Berggren
,
M.
, and
Henningson
,
D. S.
,
1999
, “
Optimal Disturbances and Bypass Transition in Boundary Layers
,”
Phys. Fluids
,
11
, pp.
134
150
.10.1063/1.869908
10.
Luchini
,
P.
,
2000
, “
Reynolds-Number Independent Instability of the Boundary Layer Over a Flat Surface. Part 2: Optimal Perturbations
,”
J. Fluid Mech.
,
404
, pp.
289
309
.10.1017/S0022112099007259
11.
Pralits
,
J. O.
,
Airiau
,
C.
,
Hanifi
,
A.
, and
Henningson
,
D. S.
,
2000
, “
Sensitivity Analysis Using Adjoint Parabolized Stability Equations for Compressible Flows
,”
Flow Turbul. Combust.
,
65
, pp.
321
346
.10.1023/A:1011434805046
12.
Luchini
,
P.
, and
Bottaro
,
A.
,
2014
, “
Adjoint Equations in Stability Analysis
,”
Ann. Rev. Fluid Mech.
,
46
, pp.
493
517
.10.1146/annurev-fluid-010313-141253
13.
Giannetti
,
F.
, and
Luchini
,
P.
,
2007
, “
Structural Sensitivity of the First Instability of the Cylinder Wake
,”
J. Fluid Mech.
,
581
, pp.
167
197
.10.1017/S0022112007005654
14.
Bottaro
,
A.
,
Corbett
,
P.
, and
Luchini
,
P.
,
2003
, “
The Effect of Base Flow Variation on Flow Stability
,”
J. Fluid Mech.
,
476
, pp.
293
302
.10.1017/S002211200200318X
15.
Marquet
,
O.
,
Sipp
,
D.
, and
Jacquin
,
L.
,
2008
, “
Sensitivity Analysis and Passive Control of Cylinder Flow
,”
J. Fluid Mech.
,
615
, pp.
221
252
.10.1017/S0022112008003662
16.
Pralits
,
J. O.
,
Brandt
,
L.
, and
Giannetti
,
F.
,
2010
, “
Instability and Sensitivity of the Flow Around a Rotating Circular Cylinder
,”
J. Fluid Mech.
,
650
, pp.
513
536
.10.1017/S0022112009993764
17.
Strykowski
,
P. J.
, and
Sreenivasan
,
K. R.
,
1990
, “
On the Formation and Suppression of Vortex ‘Shedding’ at Low Reynolds Number
,”
J. Fluid Mech.
,
218
, pp.
71
107
.10.1017/S0022112090000933
18.
Sipp
,
D.
,
Marquet
,
O.
,
Meliga
,
P.
, and
Barbagallo
,
A.
,
2010
, “
Dynamics and Control of Global Instabilities in Open Flows: A Linearized Approach
,”
ASME Appl. Mech. Rev.
,
63
(3), p.
030801
.10.1115/1.4001478
19.
Theofilis
,
V.
,
2011
, “
Global Linear Instability
,”
Ann. Rev. Fluid Mech.
,
43
, pp.
319
352
.10.1146/annurev-fluid-122109-160705
20.
Brandt
,
L.
,
Sipp
,
D.
,
Pralits
,
J. O.
, and
Marquet
,
O.
,
2011
, “
Effect of Base-Flow Variation in Noise Amplifiers: The Flat-Plate Boundary Layer
,”
J. Fluid Mech.
,
687
, pp.
503
528
.10.1017/jfm.2011.382
21.
Åkervik
,
E.
,
Brandt
,
L.
,
Henningson
,
D. S.
,
Hœpffner
,
J.
,
Marxen
,
O.
, and
Schlatter
,
P.
,
2006
, “
Steady Solutions of the Navier–Stokes Equations by Selective Frequency Damping
,”
Phys. Fluids
,
18
, p.
068102
.10.1063/1.2211705
22.
Chomaz
,
J. M.
,
2005
, “
Global Instabilities in Spatially Developing Flows: Non-Normality and Nonlinearity
,”
Ann. Rev. Fluid Mech.
,
37
, pp.
357
392
.10.1146/annurev.fluid.37.061903.175810
23.
Trefethen
,
L. N.
, and
Bau
,
D.
,
1997
,
Numerical Linear Algebra
,
SIAM
,
Philadelphia, PA
.
24.
Mack
,
C. J.
,
Schmid
,
P. J.
, and
Sesterhenn
,
J. L.
,
2008
, “
Global Stability of Swept Flow Around a Parabolic Body: Connecting Attachment-Line and Crossflow Modes
,”
J. Fluid Mech.
,
611
, pp.
205
214
.10.1017/S0022112008002851
25.
Mack
,
C. J.
, and
Schmid
,
P. J.
,
2011
, “
Global Stability of Swept Flow Around a Parabolic Body: Features of the Global Spectrum
,”
J. Fluid Mech.
,
669
, pp.
375
396
.10.1017/S0022112010005252
26.
Bagheri
,
S.
,
Schlatter
,
P.
,
Schmid
,
P. J.
, and
Henningson
,
D. S.
,
2009
, “
Global Stability of a Jet in Crossflow
,”
J. Fluid Mech.
,
624
, pp.
33
44
.10.1017/S0022112009006053
27.
Gunzburger
,
M. D.
,
2003
,
Perspectives in Flow Control and Optimization
,
SIAM
,
Philadelphia, PA
.
28.
Pringle
,
C. C. T.
, and
Kerswell
,
R. R.
,
2010
, “
Using Nonlinear Transient Growth to Construct the Minimal Seed for Shear Flow Turbulence
,”
Phys. Rev. Lett.
,
105
, p.
154502
.10.1103/PhysRevLett.105.154502
29.
Cherubini
,
S.
,
De Palma
,
P.
,
Robinet
,
J.-C.
, and
Bottaro
,
A.
,
2010
, “
Rapid Path to Transition Via Nonlinear Localized Optimal Perturbations in a Boundary-Layer Flow
,”
Phys. Rev. E
,
82
, p.
066302
.10.1103/PhysRevE.82.066302
30.
Monokrousos
,
A.
,
Bottaro
,
A.
,
Brandt
,
L.
,
Vita
,
A. D.
, and
Henningson
,
D. S.
,
2011
, “
Non-Equilibrium Thermodynamics and the Optimal Path to Turbulence in Shear Flows
,”
Phys. Rev. Lett.
,
106
, p.
134502
.10.1103/PhysRevLett.106.134502
You do not currently have access to this content.