×

zbMATH — the first resource for mathematics

Direct optimal growth analysis for timesteppers. (English) Zbl 1144.76044
Summary: Methods are described for transient growth analysis of flows with arbitrary geometric complexity, where in particular the flow is not required to vary slowly in the streamwise direction. Emphasis is on capturing the global effects arising from localized convective stability in streamwise-varying flows. The methods employ the ‘timestepper’s approach’ in which a nonlinear Navier-Stokes code is modified to provide evolution operators for both the forward and adjoint linearized equations. First, the underlying mathematical treatment in primitive flow variables is presented. Then, details are given for the inner level code modifications and outer level eigenvalue and SVD algorithms in the timestepper’s approach. Finally, some examples are shown, and guidance is provided on practical aspects of this type of large-scale stability analysis.

MSC:
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76D05 Navier-Stokes equations for incompressible viscous fluids
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65F10 Iterative numerical methods for linear systems
Software:
ARPACK; LAPACK
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Tuckerman, Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems pp 453– (2000) · doi:10.1007/978-1-4612-1208-9_20
[2] Barkley, Three-dimensional instability in flow over a backward-facing step, Journal of Fluid Mechanics 473 pp 167– (2002) · Zbl 1026.76019
[3] Barkley, Three-dimensional Floquet stability analysis of the wake of a circular cylinder, Journal of Fluid Mechanics 322 pp 215– (1996) · Zbl 0882.76028
[4] Blackburn, Three-dimensional instability and state selection in an oscillatory axisymmetric swirling flow, Physics of Fluids 14 (11) pp 3983– (2002) · Zbl 1185.76054
[5] Blackburn, The onset of three-dimensional standing and modulated travelling waves in a periodically driven cavity flow, Journal of Fluid Mechanics 497 pp 289– (2003) · Zbl 1068.76037
[6] Blackburn, Instability modes and transition of pulsatile stenotic flow: pulse-period dependence, Journal of Fluid Mechanics 573 pp 57– (2007) · Zbl 1119.76402
[7] Elston, The primary and secondary instabilities of flow generated by an oscillating circular cylinder, Journal of Fluid Mechanics 550 pp 359– (2006) · Zbl 1222.76052
[8] Sherwin, Three-dimensional instabilities and transition of steady and pulsatile flows in an axisymmetric stenotic tube, Journal of Fluid Mechanics 533 pp 297– (2005) · Zbl 1074.76021
[9] Ă…kervik, Optimal growth, model reduction and control in a separated boundary-layer flow using global eigenmodes, Journal of Fluid Mechanics 579 pp 305– (2007) · Zbl 1175.76049
[10] Blackburn, Convective instability and transient growth in flow over a backward-facing step, Journal of Fluid Mechanics 603 pp 271– (2008) · Zbl 1151.76470
[11] Ehrenstein, On two-dimensional temporal modes in spatially evolving open flows: the flat-plate boundary layer, Journal of Fluid Mechanics 536 pp 209– (2005) · Zbl 1073.76027
[12] Marquet O, Sipp D, Jacquin L. Global optimal perturbations in a separated flow over a backward-rounded-step. Thirty-sixth AIAA Fluid Dynamics Conference and Exhibit, San Francisco, 2006; Paper No. 2006-2879.
[13] Karniadakis, Spectral/hp Element Methods for Computational Fluid Dynamics (2005) · Zbl 1116.76002 · doi:10.1093/acprof:oso/9780198528692.001.0001
[14] Cossu, Global measures of local convective instabilities, Physical Review Letters 78 pp 4387– (1997)
[15] Schmid, Stability and Transition in Shear Flows (2001) · Zbl 0966.76003 · doi:10.1007/978-1-4613-0185-1
[16] Trefethen, Hydrodynamic stability without eigenvalues, Science 261 pp 578– (1993) · Zbl 1226.76013
[17] Karniadakis, High-order splitting methods for the incompressible Navier–Stokes equations, Journal of Computational Physics 97 (2) pp 414– (1991) · Zbl 0738.76050
[18] Guermond, Velocity-correction projection methods for incompressible flows, SIAM Journal on Numerical Analysis 41 pp 112– (2003) · Zbl 1130.76395
[19] Wilhelm, Stability analysis for different formulations of the nonlinear term in PN–PN-2 spectral element discretizations of the Navier–Stokes equations, Journal of Computational Physics 174 pp 306– (2001)
[20] Barkley, Confined three-dimensional stability analysis of the cylinder wake, Physical Review E 71 pp 017301– (2007)
[21] Saad, Numerical Methods for Large Eigenvalue Problems (1992) · Zbl 1242.65068
[22] Anderson, LAPACK User’s Guide (1999) · Zbl 0934.65030 · doi:10.1137/1.9780898719604
[23] Lehoucq, ARPACK Users’ Guide (1998) · Zbl 0901.65021 · doi:10.1137/1.9780898719628
[24] Kaiktsis, Onset of three-dimensionality, equilibria, and early transition in flow over a backward-facing step, Journal of Fluid Mechanics 231 pp 501– (1991) · Zbl 0728.76057
[25] Blackburn, Convective instability and transient growth in steady and pulsatile stenotic flows, Journal of Fluid Mechanics 603 pp 271– (2008) · Zbl 1145.76374
[26] Butler, Three-dimensional optimal perturbations in viscous shear flow, Physics of Fluids A 4 (8) pp 1637– (1992)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.