Abstract
The perturbations triggered by free-stream vortical disturbances in compressible boundary layers developing over concave surfaces are investigated by asymptotic methods and numerically. We utilize an asymptotic framework based on the limits of high Reynolds and Görtler numbers and solve the receptivity problem by integrating the initial-boundary value problem coupled with appropriate initial and boundary conditions, synthesizing the effect of the free-stream vortical disturbances. The conditions for which the Görtler vortices start to grow are revealed and we find that the Mach number is destabilizing when the spanwise diffusion is negligible and stabilizing when the boundary-layer thickness is comparable with the spanwise wavelength of the vortices. During the initial development of the vortices, only the receptivity calculations are accurate. At streamwise locations where the free-stream disturbances have fully decayed, the growth rate and wavelength are computed with sufficient accuracy by an eigenvalue method, although the correct amplitude and shape of the Görtler vortices can only be determined by the receptivity calculations. We find that Klebanoff modes always evolve from the leading edge, the Görtler vortices dominate when the influence of the curvature becomes significant, and highly-oblique Tollmien-Schlichting waves may precede the Görtler vortices for moderate Görtler numbers. We also obtain the neutral stability curves, i.e. curves that distinguish spatially growing from spatially decaying perturbations at different streamwise locations. For relatively high frequencies the triple-deck formalism allows us to confirm the numerical results of the negligible influence of the curvature on the Tollmien-Schlichting waves when the Görtler number is an order-one quantity. Experimental data for compressible Görtler flows are mapped onto our neutral-curve graphs and earlier theoretical results are compared with our predictions.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Hall, P.: Görtler vortices in growing boundary layers: the leading edge receptivity problem, linear growth and the nonlinear breakdown stage. Mathematika 37(74), 151–189 (1990)
Saric, W.: Görtler vortices. Annu. Rev. Fluid Mech. 26(1), 379–409 (1994)
Mayle, R.: The role of laminar-turbulent transition in gas turbine engines. J. Turbom. 113(4), 509–537 (1991)
Volino, R., Simon, T.: Bypass transition in boundary layers including curvature and favorable pressure gradient effects. J. Turbomach. 117(1), 166–174 (1995)
Mangalam, S., Dagenhart, J., Hepner, T., Meyers, J.: The Görtler instability on an airfoil. AIAA Paper, pp. 85–0491 (1985)
Chen, F., Malik, M., Beckwith, I.: Görtler instability and supersonic quiet nozzle design. AIAA J. 30(8), 2093–2094 (1992)
Wu, X., Zhao, D., Luo, J.: Excitation of steady and unsteady Görtler vortices by free-stream vortical disturbances. J. Fluid Mech. 682, 66–100 (2011)
Ricco, P., Wu, X.: Response of a compressible laminar boundary layer to free-stream vortical disturbances. J. Fluid Mech. 587, 97–138 (2007)
Viaro, S. Ricco, P.: Neutral stability curves of low-frequency Görtler flow generated by free-stream vortical disturbances. J. Fluid Mech. 845, no. R1 (2018)
Viaro, S., Ricco, P.: Compressible unsteady Görtler vortices subject to free-stream vortical disturbances. J. Fluid Mech. 867, 250–299 (2019)
Viaro, S., Ricco, P.: Neutral stability curves of compressible Görtler flow generated by low-frequency free-stream vortical disturbances. J. Fluid Mech. 876, 1146–1157 (2019)
Leib, S., Wundrow, D., Goldstein, M.: Effect of free-stream turbulence and other vortical disturbances on a laminar boundary layer. J. Fluid Mech. 380, 169–203 (1999)
El-Hady, N., Verma, A.: Growth of Görtler vortices in compressible boundary layers along curved surfaces. J. Eng. Appl. Sci. 2(3), 213–238 (1983)
Ciolkosz, L., Spina, E.: An experimental study of Görtler vortices in compressible flow. AIAA Paper 4512, 1–21 (2006)
Acknowledgements
We wish to acknowledge the financial support of the US Air Force through the AFOSR grant FA9550-15-1-0248 with Prof. Russell Cummings as the Program Officer. We also thank Dr Marvin E. Goldstein for the invaluable discussions about this research work.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Ricco, P., Viaro, S. (2022). Receptivity of Unsteady Compressible Görtler Flows to Free-Stream Vortical Disturbances. In: Sherwin, S., Schmid, P., Wu, X. (eds) IUTAM Laminar-Turbulent Transition. IUTAM Bookseries, vol 38. Springer, Cham. https://doi.org/10.1007/978-3-030-67902-6_68
Download citation
DOI: https://doi.org/10.1007/978-3-030-67902-6_68
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-67901-9
Online ISBN: 978-3-030-67902-6
eBook Packages: EngineeringEngineering (R0)