Abstract
The Chebyshev tau method is examined in detail for a variety of eigenvalue problems arising in hydrodynamic stability studies, particularly those of Orr-Sommerfeld type. We concentrate on determining the whole of the top end of the spectrum in parameter ranges beyond those often explored. The method employing a Chebyshev representation of the fourth derivative operator, D4, is compared with those involving the second and first derivative operators, D2 and D, respectively. The latter two representations require use of the QZ algorithm in the resolution of the singular generalised matrix eigenvalue problem which arises. Physical problems explored are those of Poiseuille flow, Couette flow, pressure gradient driven circular pipe flow, and Couette and Poiseuille problems for two viscous, immiscible fluids, one overlying the other.
| Original language | English |
|---|---|
| Pages (from-to) | 399-434 |
| Number of pages | 36 |
| Journal | Applied Numerical Mathematics |
| Volume | 22 |
| Issue number | 4 |
| DOIs | |
| State | Published - Dec 1996 |
Funding
This research was supported in part by an appointment to the postgraduate research program at the Oak Ridge National Laboratory (ORNL) administered by the Oak Ridge Institute for Science and Education. We are very grateful to three anonymous referees for their careful reading of the paper and trenchant suggestions for improvement. There has been much recent attention directed at solving difficult eigenvalue problems for differential equations like the Orr-Sommerfeld one, with particular interest in the removal of spurious eigenvalues or calculations in high Reynolds number ranges, cf. \[1 ,7,13,14,17-19,22,30\]. Equations of Orr-Sommerfeld type govern the stability of shear and related flows which have important applications in many fields. One such field is climate modelling with questions like determining an explanation for the origin of the mid-latitude cyclone which in turn is responsible for producing the high and This work was supported in part by the Office of Scientific Computing, U.S. Department of Energy under contract DE-AC05-84OR21400 with Lockheed Martin Energy Systems, and by the Advanced Research Projects Agency under contract DAAL03-91-C-0047, administered by the Army Research Office. * Corresponding author. E-mail: [email protected]. LP ermanent address: Department of Mathematics, The University, Glasgow, G12 8QW, UK.
| Funders | Funder number |
|---|---|
| Lockheed Martin Energy Systems | |
| Office of Scientific Computing | |
| U.S. Department of Energy | DE-AC05-84OR21400 |
| Army Research Office | |
| Oak Ridge National Laboratory | |
| Oak Ridge Institute for Science and Education | |
| Advanced Research Projects Agency | DAAL03-91-C-0047 |
Keywords
- Chebyshev polynomials
- Eigenvalue problems
- Multilayer flows
- Orr-sommerfeld equations
- Qz algorithm