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Investigation of the stability of boundary layers by a finite-difference model of the Navier—Stokes equations

Published online by Cambridge University Press:  11 April 2006

H. Fasel
H. Fasel
Affiliation:
Institut A für Mechanik, Universität Stuttgart, Germany
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Abstract

The stability of incompressible boundary-layer flows on a semi-infinite flat plate and the growth of disturbances in such flows are investigated by numerical integration of the complete Navier–;Stokes equations for laminar two-dimensional flows. Forced time-dependent disturbances are introduced into the flow field and the reaction of the flow to such disturbances is studied by directly solving the Navier–Stokes equations using a finite-difference method. An implicit finitedifference scheme was developed for the calculation of the extremely unsteady flow fields which arose from the forced time-dependent disturbances. The problem of the numerical stability of the method called for special attention in order to avoid possible distortions of the results caused by the interaction of unstable numerical oscillations with physically meaningful perturbations. A demonstration of the suitability of the numerical method for the investigation of stability and the initial growth of disturbances is presented for small periodic perturbations. For this particular case the numerical results can be compared with linear stability theory and experimental measurements. In this paper a number of numerical calculations for small periodic disturbances are discussed in detail. The results are generally in fairly close agreement with linear stability theory or experimental measurements.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 78 , Issue 2 , 23 November 1976 , pp. 355 - 383
Copyright
© 1976 Cambridge University Press

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References

Barnes, F. H. 1966 A hot wire anemometer study of the effect of disturbances on the laminar boundary layer on a flat plate. Ph.D. thesis, University of Edinburgh.
Barry, M. D. J. & Ross, M. A. S. 1970 The flat plate boundary layer. Part 2. The effect of increasing thickness on stability J. Fluid Mech. 43, 813–818.Google Scholar
Benney, D. J. & Lin, C. C. 1960 On the secondary motion induced by oscillations in a shear flow Phys. Fluids, 3, 656–657.Google Scholar
Bouthier, M. 1972, 1973 Stabilité linéaire des écoulements presque parallèles. J. Méc. 11, 599–621, 12, 75–95.Google Scholar
Cheng, S. I. 1970 Numerical integration of Navier—Stokes equations A.I.A.A. J. 8, 2115–2122.Google Scholar
Fasel, H. 1974 Untersuchungen zum Problem des Grenzschichtumschlages durch numerische Integration der Navier—Stokes Gleichungen. Dissertation, University of Stuttgart.
Gaster, M. 1974 On the effects of boundary-layer growth on flow stability. J. Fluid Mech. 66, 465–480.Google Scholar
Greenspan, H. P. & Benney, D. J. 1963 On shear-layer instability, breakdown and transition J. Fluid Mech. 15, 133–153.Google Scholar
Hadamard, J. 1952 Lectures on Cauchy's Problem in Linear Partial Differential Equations. Dover.
Jordinson, R. 1968 The transition from laminar to turbulent flow over a flat plate-space amplified, numerical solutions of the Orr—Sommerfeld equation. Ph.D. thesis, University of Edinburgh.
Jordinson, R. 1970 The flat plate boundary layer. Part 1. Numerical integration of the Orr—Sommerfeld equation J. Fluid Mech. 43, 801–811.Google Scholar
Kümmerer, H. 1973 Numerische Untersuchungen zur Stabilität ebener laminarer Grenzschichtströmungen. Dissertation, University of Stuttgart.
Richtmyer, R. D. & Morton, K. W. 1967 Difference Methods for Initial-Value Problems. Interscience.
Roache, P. J. 1972 On artificial viscosity J. Comp. Phys. 10, 169–184.Google Scholar
Ross, J. A., Barnes, F. H., Burns, J. G. & Ross, M. A. S. 1970 The flat plate boundary layer. Part 3. Comparison of theory with experiment J. Fluid Mech. 43, 819–832.Google Scholar
Schlichting, H. 1965 Grenzschicht-Theorie, 5th edn. Karlsruhe: G. Braun.
Schubauter, G. B. & Skramstad, H. K. 1948 Laminar boundary-layer oscillations and transition on a flat plate. N.A.C.A. Rep. no. 909.Google Scholar
Stuart, J. T. 1965 Hydrodynamic stability Appl. Mech. Rev. 18, 523–531.Google Scholar
Thomas, L. H. 1949 Elliptic problems in linear difference equations over a network. Watson Sci. Comp. Lab. Rep., Columbia Univ., New York.Google Scholar
Wazzan, A. R., Okamura, T. T. & Smith, A. M. O. 1968 Spatial and temporal stability charts for the Falkner—Skan boundary-layer profiles. McDonnell-Douglas Astronautics Co., Rep. no. DAC-67086.Google Scholar