Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-25T19:05:18.105Z Has data issue: false hasContentIssue false
  • English
  • Français

The stability of a curved, heated boundary layer: linear and nonlinear problems

Published online by Cambridge University Press:  17 February 2009

C. E. Watson  and
S. R. Otto
C. E. Watson
Affiliation:
Quintessa Limited, Dalton House, Newtown Road, Henley-On-Thames, Oxfordshire, RG9 IHG, England.
S. R. Otto
Affiliation:
R&A Rules Limited, Beach House, Golf Place, St Andrews, KY16 9JA, Scotland; e-mail: [email protected].
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider the stability of high Reynolds number flow past a heated, curved wall. The influence of both buoyancy and curvature, with the appropriate sense, can render a flow unstable to longitudinal vortices. However, conversely each mechanism can make a flow more stable; as with a stable stratification or a convex curvature. This is partially due to their influence on the basic flow and also due to additional terms in the stability equations. In fact the presence of buoyancy in combination with an appropriate local wall gradient can actually increase the wall shear and these effects can lead to supervelocities and the promotion of a wall jet. This leads to the interesting discovery that the flow can be unstable for both concave and convex curvatures. Furthermore, it is possible to observe sustained vortex growth in stably stratified boundary layers over convexly curved walls. The evolution of the modes is considered in both the linear and nonlinear régimes.

Type
Research Article
Information
The ANZIAM Journal , Volume 46 , Issue 4 , April 2005 , pp. 507 - 543
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Benmalek, A. and Saric, W. S., “Effects of curvature variations on the nonlinear evolution of Goertler vortices”, Phys. Fluids 6 (1994) 3353–3367.CrossRefGoogle Scholar
[2]Blasius, H., “Grenzschichten in flüssigkeiten mit kleiner Reibung”, Z Math. Phys. 56 (1908) 1–37.Google Scholar
[3]Cebeci, T. and Bradshaw, P., Momentum transfer in boundary layers (Hemisphere, London, 1977).Google Scholar
[4]Cole, T. R., Otto, S. R. and Watson, C. E., “The nonlinear growth of Görtler vortices in curved mixing layers and their effect on the inherent inviscid modes”, 2005, (in preparation).Google Scholar
[5]Denier, J. P. and Hall, P., “On the nonlinear development of the most unstable Görtler vortex mode”, J. Fluid Mech. 247 (1993) 1–16.CrossRefGoogle Scholar
[6]Denier, J. P., Hall, P. and Seddougui, S. O., “On the receptivity problem for Görtler vortices: vortex motions induced by wall roughness”, Philos. Trans. Roy. Soc. London A 335 (1991) 51–85.Google Scholar
[7]G, H.örtler, “On the three-dimensional instability of laminar boundary layers on concave walls”, NACA Tech. Mem. (1940) 1375.Google Scholar
[8]Hall, P., “Taylor-Görtler vortices in fully developed or boundary layer flows: linear theory”, J. Fluid Mech. 124 (1982) 475–494.CrossRefGoogle Scholar
[9]Hall, P., “The linear development of Görtler vortices in growing boundary layers”, J. Fluid Mech. 130 (1983) 41–58.CrossRefGoogle Scholar
[10]Hall, P., “The nonlinear development of Görtler vortices in growing boundary layers”, J. Fluid Mech. 193 (1988) 243–266.CrossRefGoogle Scholar
[11]Hall, P., “Görtler vortices in growing boundary layers: the leading edge receptivity problem, linear growth and the nonlinear breakdown stage”, Mathematika 37 (74) (1990) 151–189.CrossRefGoogle Scholar
[12]Hall, P., “Streamwise vortices in heated boundary layers”, J. Fluid Mech. 252 (1993) 301–324.CrossRefGoogle Scholar
[13]Hall, P. and Lakin, W. D., “The fully nonlinear development of Görtler vortices in growing boundary layers”, Proc. Roy. Soc. London A 415 (1988) 421–444.Google Scholar
[14]Hall, P. and Morris, H., “On the instability of boundary layers on heated flat plates”, J. Fluid Mech. 245 (1992) 367–400.CrossRefGoogle Scholar
[15]Itoh, N., “A non-parallel theory for Göler instability of Falkner-Skan boundary layers”, Fluid Dyn. Res. 28 (2001) 383–396.CrossRefGoogle Scholar
[16]Lee, K. and Liu, J. T. C., “On the growth of mushroomlike structures in nonlinear spatially developing Goertler vortex flow”, Phys.Fluids A 4 (1992) 95–103.CrossRefGoogle Scholar
[17]Mangalam, S. M., Dagenhart, J. R., Hepner, T. E. and Meyers, J. F., “The Görtler instability on an airfoil”, 23rd Aerodynamic Sciences Meeting, January 14–17, 1985, AIAA-85–0491.CrossRefGoogle Scholar
[18]Otto, S. R., Stott, J. A. K. and Denier, J. P., “On the role of buoyancy in determining the stability of curved mixing layers”, Phys. Fluids 11 (6) (1999) 1495–1501.CrossRefGoogle Scholar
[19]Owen, D. J., Seddougui, S. O. and Otto, S. R., “The linear evolution of centrifugal instabilities in curved, compressible mixing layers”, Phys. Fluids 9 (1997) 2506–2518.CrossRefGoogle Scholar
[20]Sarkies, J. M. and Otto, S. R., “Görtler vortices in compressible mixing layers”, J. Fluid Mech. 427 (2001) 359–388.CrossRefGoogle Scholar
[21]Scorer, R. S., Dynamics of meteorology and climate, Wiley Praxis Series in Atmospheric Physics (John Wiley & Sons, Chichester, 1997).Google Scholar
[22]Spiegel, E. A. and Veronis, G., “On the Boussinesq approximlation for a compressible fluid”, J. Astrophys. 131 (1960) 442–447.CrossRefGoogle Scholar
[23]Stott, J. A. K. and Denier, J. P., “The stability of boundary layers on curved heated plates”, ANZIAM J. 43 (2002) 333–358.CrossRefGoogle Scholar
[24]Swearingen, J. D. and Blackwelder, R. F., “The growth and breakdown of streaniwise vortices in the presence of a wall”, J. Fluid Mech. 182 (1987) 255.CrossRefGoogle Scholar
[25]Watson, C. E., Otto, S. R. and Jackson, T. L., “The receptivity and evolution of longitudinal vortices: the interplay between buoyancy and curvature within mixing layers”, 2005, submitted to Theor. Comp. Fluid Dynamics.Google Scholar
[26]Wu, R. S. and Cheng, K. C., “Thermal instability of Blasius flow along horizontal plates”, Intl. J. Heat Mass Transfer 105 (1976) 907–913.CrossRefGoogle Scholar