The parabolised stability equations for 3D-flows: implementation and numerical stability

The parabolised stability equations for 3D-flows: implementation and numerical stability

The numerical implementation of the parabolised stability equations (PSE) using a spectral/ hp-element discretisation is considered, and the numerical stability of the governing equations is presented. Analogous to the primitive variable form of the two-dimensional PSE, the equations are ill-posed;...

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Bibliographic Details
Published in:Applied numerical mathematics Vol. 58; no. 7; pp. 1017 - 1029
Main Authors: Broadhurst, Michael S., Sherwin, Spencer J.
Format: Journal Article Conference Proceeding
Language:English
Published: Amsterdam Elsevier B.V 01-07-2008
Elsevier
Subjects:
Exact sciences and technology
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Numerical stability
PSE
Sciences and techniques of general use
Spectral/ hp-methods
Static condensation
Static condensation
Spectral/ hp-methods
PSE
Numerical stability
Three-dimensional calculations
Acoustic wave
Spectral/hp-methods
Acoustics
Numerical method
Euler scheme
Implementation
Two dimensional equation
Numerical analysis
Applied mathematics
Two-dimensional calculations
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Summary:The numerical implementation of the parabolised stability equations (PSE) using a spectral/ hp-element discretisation is considered, and the numerical stability of the governing equations is presented. Analogous to the primitive variable form of the two-dimensional PSE, the equations are ill-posed; although choosing an Euler implicit scheme in the streamwise z-direction yields a stable scheme for sufficiently large step sizes ( Δ z > 1 / | β | , where β is the streamwise wavenumber). The source of the instability is a residual ellipticity that remains in the equations, and presents itself as an upstream propagating acoustic wave. Neglecting this term relaxes the lower limit on the step-size restriction. The θ-scheme is also considered, allowing the step-size restriction of the scheme to be determined. The explicit scheme is always unstable, whereas neglecting the pressure gradient term shows stable eigenspectra for θ ⩾ 0.5 .
ISSN:0168-9274
1873-5460
DOI:10.1016/j.apnum.2007.04.016