Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016 pp 639-650 | Cite as

# Three-Dimensional Flow Stability Analysis Based on the Matrix-Forming Approach Made Affordable

## Abstract

Theoretical developments for hydrodynamic instability analysis are often based on eigenvalue problems, the size of which depends on the dimensionality of the reference state (or base flow) and the number of coupled equations governing the fluid motion. The straightforward numerical approach consisting on spatial discretization of the linear operators, and numerical solution of the resulting matrix eigenvalue problem, can be applied today without restrictions to one-dimensional base flows. The most efficient implementations for one-dimensional problems feature spectral collocation discretizations which produce dense matrices. However, this combination of theoretical approach and numerics becomes computationally prohibitive when two-dimensional and three-dimensional flows are considered. This paper proposes a new methodology based on an optimized combination of high-order finite differences and sparse algebra, that leads to a substantial reduction of the computational cost. As a result, three-dimensional eigenvalue problems can be solved in a local workstation, while other related theoretical methods based on the WKB expansion, like global-oscillator instability or the Parabolized Stability Equations, can be extended to three-dimensional base flows and solved using a personal computer.

## Notes

### Acknowledgements

The work of D.R. was funded by the Brazilian Science without borders/CAPES “Attraction of Young Talents” Fellowship, grant 88881.064930/2014-01. This work is also supported by the São Paulo State Research Foundation (FAPESP) grants 2014/24782-0 and 2017/01586-0 and Brazilian National Counsel of Technological and Scientific Development (CNPq) grants 423846/2016-7, 405144/2016-4 and 305512/2016-1.

## References

- 1.P.R. Amestoy, I.S. Duff, J.Y. L’xcellent, J. Koster, A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Appl.
**23**(1), 15–41 (2001)CrossRefzbMATHMathSciNetGoogle Scholar - 2.W.E. Arnoldi, The principle of minimized iterations in the solution of the matrix eigenvalue problem. Q. Appl. Math.
**9**, 17–29 (1951)CrossRefzbMATHMathSciNetGoogle Scholar - 3.M. Broadhurst, S. Sherwin, The parabolised stability equations for 3D-flows: implementation and numerical stability. Appl. Numer. Math.
**58**(7), 1017–1029 (2008)CrossRefzbMATHMathSciNetGoogle Scholar - 4.S. Chiba, Global stability analysis of incompressible viscous flow. J. Jpn. Soc. Comput. Fluid Dyn.
**7**, 20–48 (1998)Google Scholar - 5.E.M. Gennaro, D. Rodríguez, M.A.F. de Medeiros, V. Theofilis, Sparse techniques in global flow instability with application to compressible leading-edge flow. AIAA J.
**51**(9), 2295–2303 (2013)CrossRefGoogle Scholar - 6.G.H. Golub, C.F. Van Loan,
*Matrix Computations*(John Hopkins University Press, Baltimore, 1989)zbMATHGoogle Scholar - 7.T. Herbert, Parabolized stability equations. Annu. Rev. Fluid Mech.
**29**, 245–283 (1997)CrossRefMathSciNetGoogle Scholar - 8.P. Huerre, P. Monkewitz, Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech.
**22**, 473–537 (1990)CrossRefzbMATHMathSciNetGoogle Scholar - 9.S.A. Orszag, Accurate solution of the Orr-Sommerfeld stability equation. J. Fluid Mech.
**5**(4), 689–703 (1971)CrossRefzbMATHGoogle Scholar - 10.R.T. Pierrehumbert, S.E. Widnall, The two- and three-dimensional instabilities of spatially periodic shear layer. J. Fluid Mech.
**114**, 59–82 (1982)CrossRefzbMATHGoogle Scholar - 11.D. Rodríguez, E.M. Gennaro, On the secondary instability of forced and unforced laminar separation bubbles. Procedia IUTAM
**14**, 78–87 (2015)CrossRefGoogle Scholar - 12.D. Rodríguez, V. Theofilis, Massively parallel numerical solution of the biglobal linear instability eigenvalue problem using dense linear algebra. AIAA J.
**47**(10), 2449–2459 (2009)CrossRefGoogle Scholar - 13.D. Rodríguez, A. Tumin, V. Theofilis, Towards the foundation of a global mode concept, in
*6th AIAA Theoretical Fluid Mechanics Conference, 27–30 June, Honolulu, USA*(AIAA, 2011), pp. 2011–3603.Google Scholar - 14.D. Rodríguez, E.M. Gennaro, M.P. Juniper, The two classes of primary modal instability in laminar separation bubbles. J. Fluid Mech.
**734**, R4 (2013)CrossRefzbMATHGoogle Scholar - 15.P.J. Schmid, D.S. Henningson,
*Stability and Transition in Shear Flows*(Springer, New York, 2001)CrossRefzbMATHGoogle Scholar - 16.S. Sherwin, H. Blackburn, Three-dimensional instabilities and transition of steady and pulsatile axisymmetric. J. Fluid Mech.
**533**, 297–327 (2005)CrossRefzbMATHMathSciNetGoogle Scholar - 17.V. Theofilis, Global linear stability. Annu. Rev. Fluid Mech.
**43**, 319–352 (2011)CrossRefzbMATHMathSciNetGoogle Scholar - 18.A. Zebib, Instabilities of viscous flow past a circular cylinder. J. Eng. Math.
**21**, 155–165 (1987)CrossRefzbMATHGoogle Scholar