Abstract
An auto-adaptive multidomain pseudo-spectral technique is considered in order to solve the linear stability problem of viscous compressible flows. Both the locations of the interfaces and the parameters of the mappings in each subdomain are adapted by minimizing the H 2 ω -norm of the calculated solution. Such method provides automatically—this is the key point—the best polynomial interpolation of the basic state the stability of which is studied. It turns out that the whole procedure is needed to obtain reliable results. The method is first validated against results available in the literature (both viscous incompressible and inviscid compressible Rayleigh–Taylor configurations). The efficiency of the numerical method is illustrated with results on the linear stability of the compressible viscous diffusive Rayleigh–Taylor flow where no analytical or numerical results are available. New results showing the influence of stratification, viscosity, diffusity between species and thermal diffusivity are presented.
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REFERENCES
Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. A. (1988). Spectral Methods in Fluids Dynamics, Springer-Verlag, New-York.
Serre, E., Crespo, E., and Bontoux, P. (2001). Annular and spiral patterns in flows between rotating and stationary disks. J. Fluid Mech. 434, 65–100.
Serre, E., and Pulicani, J.-P. (2001). A 3D pseudospectral method for convection in a rotating cylinder. Comput. & Fluids 30(4), 491–519.
Gauthier, S., Guillard, H., Lumpp, T., Malé, J.-M., Peyret, R., and Renaud, F. (1996). A Spectral Domain Decomposition Technique with Moving Interfaces for Viscous Compressible Flows, Proceedings of the ECCOMAS 96, Paris, pp. 839–843.
Orszag, S. A., and Patera, A. T. (1983). Secondary instability of wall-bounded shear flows. J. Fluid Mech. 128, 347–385.
Gauthier, S., Gamess, A., et Iooss, G. (1990). Chaotic behavior in oscillatory compressible convection in extended boxes for small Prandtl numbers. Europhys. Lett. 13, 117–122.
Renaud, F., and Gauthier, S. (1993). E´ tude de la stabilité linéaire de la couche limite compressible auto-semblable sur une plaque plane, Technical Report CEA-N-2743.
Macaraeg, M. G., and Streett, C. L. (1986). Improvements in spectral collocation discretization through a multiple domain technique. Appl. Numer. Math. 2, 95–108.
Macaraeg, M. G., and Streett, C. L. (1988). A spectral multi-domain technique for viscous compressible reacting flows. Internat. J. Numer. Methods Fluids 8, 1121–1134.
Guillard, H., Malé, J.-M., and Peyret, R. (1992). Adaptive spectral methods with application to mixing layer computation. J. Comput. Phys. 102, 114–127.
Renaud, F., and Gauthier, S. (1997). A dynamical pseudo-spectral domain decomposition technique: Application to viscous compressible flows. J. Comput. Phys. 131, 89–108.
Guillard, H., and Peyret, R. (1988). On the use of spectral methods for the numerical solution of stiff problems. Comput. Methods Appl. Mech. Eng. 66, 17–43.
Chandrasekhar, S. (1961). Hydrodynamic and Hydromagnetic Stability, Oxford, Clarendon Press.
Hirschfelder, J. O., Curtiss, C. F., and Bird, R. B. (1954). Molecular Theory of Gases and Liquids, New York, John Wiley and Sons.
Bayliss, A., and Matkowsky, B. J. (1987). Fronts, relaxation oscillations, and period doubling in solid fuel combustion. J. Comput. Phys. 71, 147–168.
Bayliss, A., Belytschoko, T., Hansen, D., and Turkel, E. (1992). Adaptive multi-domain spectral methods. In Keyes, Chan, Meurant, Scrooggs, and Voigt (eds.), Proceedings, 5th SIAM Conference on Domain Decomposition Methods for Partial Differential Equations.
Smith, B. T., Boyle, J. M., Dongarra, J. J., Garbow, B. S., Ikebe, Y., Klema, V. C., and Moler, C. B. (1976). Matrix Eigensystem Routines EISPACH, Guide. Berlin, Springer, pp. 124–125.
Mathews W. G., and Blumenthal, G. R. (1977). Rayleigh-Taylor stability of compressible and incompressible radiation-supported surfaces and slabs: application to QSO clouds. Ap. J. 214, 10–20.
Duff, R. E., Harlow, F. H., and Hirt, C. W. (1962). Effects of diffusion on interface instability between gases. Phys. Fluids 5, 417–425.
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Serre, E., Gauthier, S. An Auto-Adaptive Multidomain Spectral Technique for Linear Stability Analysis: Application to Viscous Compressible Flows. Journal of Scientific Computing 17, 153–165 (2002). https://doi.org/10.1023/A:1015196413614
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DOI: https://doi.org/10.1023/A:1015196413614