Hostname: page-component-84b7d79bbc-x5cpj Total loading time: 0 Render date: 2024-08-04T08:21:11.489Z Has data issue: false hasContentIssue false
  • English
  • Français

Compressible unsteady Görtler vortices subject to free-stream vortical disturbances

Published online by Cambridge University Press:  21 March 2019

Samuele Viaro Open the ORCID record for Samuele Viaro [Opens in a new window]  and
Pierre Ricco Open the ORCID record for Pierre Ricco [Opens in a new window]
Samuele Viaro
Affiliation:
Department of Mechanical Engineering, The University of Sheffield, SheffieldS1 3JD, UK
Pierre Ricco*
Affiliation:
Department of Mechanical Engineering, The University of Sheffield, SheffieldS1 3JD, UK
*
†Email address for correspondence: p.ricco@sheffield.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

The perturbations triggered by free-stream vortical disturbances in compressible boundary layers developing over concave walls are studied numerically and through asymptotic methods. We employ an asymptotic framework based on the limit of high Görtler number, the scaled parameter defining the centrifugal effects; we use an eigenvalue formulation where the free-stream forcing is neglected; and we solve the receptivity problem by integrating the compressible boundary-region equations complemented by appropriate initial and boundary conditions that synthesize the influence of the free-stream vortical flow. Near the leading edge, the boundary-layer perturbations develop as thermal Klebanoff modes and, when centrifugal effects become influential, these modes turn into thermal Görtler vortices, i.e. streamwise rolls characterized by intense velocity and temperature perturbations. The high-Görtler-number asymptotic analysis reveals the condition for which the Görtler vortices start to grow. 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. When the Görtler number is large, the theoretical analysis also shows that the vortices move towards the wall as the Mach number increases. These results are confirmed by the receptivity analysis, which additionally clarifies that the temperature perturbations respond to this reversed behaviour further downstream than the velocity perturbations. A matched-asymptotic composite profile, found by combining the inviscid core solution and the near-wall viscous solution, agrees well with the receptivity profile sufficiently downstream and at high Görtler number. The Görtler vortices tend to move towards the boundary-layer core when the flow is more stable, i.e. as the frequency or the Mach number increase, or when the curvature decreases. As a consequence, a region of unperturbed flow is generated near the wall. We also find that the streamwise length scale of the boundary-layer perturbations is always smaller than the free-stream streamwise wavelength. 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 the eigenvalue analysis, although the correct amplitude and evolution of the Görtler vortices can only be determined by the receptivity calculations. It is further proved that the eigenvalue predictions of the growth rate and wavenumber worsen as the Mach number increases as these quantities show a dependence on the wall-normal direction. We conclude by qualitatively comparing our results with the direct numerical simulations available in the literature.

JFM classification

Boundary Layers: Boundary layer receptivity Compressible Flows: Compressible boundary layers
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 867 , 25 May 2019 , pp. 250 - 299
Copyright
© 2019 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Boiko, A. V., Ivanov, A. V., Kachanov, Y. S. & Mischenko, D. A. 2010a Investigation of weakly-nonlinear development of unsteady Görtler vortices. Thermophys. Aeromech. 17 (4), 455–481.Google Scholar
Boiko, A. V., Ivanov, A. V., Kachanov, Y. S. & Mischenko, D. A. 2010b Steady and unsteady Görtler boundary-layer instability on concave wall. Eur. J. Mech. (B/Fluids) 29 (2), 61–83.Google Scholar
Boiko, A. V., Ivanov, A. V., Kachanov, Y. S., Mischenko, D. A. & Nechepurenko, Y. M. 2017 Excitation of unsteady Görtler vortices by localized surface nonuniformities. Theor. Comput. Fluid Dyn. 31 (1), 67–88.Google Scholar
Borodulin, V. I., Ivanov, A. V., Kachanov, Y. S. & Mischenko, D. A. 2017 Systematic study of distributed excitation of unsteady Görtler modes by freestream vortices. Eur. J. Mech. (B/Fluids) 68, 167–183.Google Scholar
Cebeci, T. 2002 Convective Heat Transfer. Springer.Google Scholar
Chen, F. J., Malik, M. R. & Beckwith, I. E. 1992 Gortler instability and supersonic quiet nozzle design. AIAA J. 30 (8), 2093–2094.Google Scholar
Choudhari, M.1996 Boundary layer receptivity to three-dimensional unsteady vortical disturbances in the free stream. AIAA Paper 96-0181.Google Scholar
Ciolkosz, L. D. & Spina, E. F.2006 An experimental study of Görtler vortices in compressible flow. AIAA Paper 4512.Google Scholar
Dando, A. H. & Seddougui, S. O. 1993 The compressible Görtler problem in two-dimensional boundary layers. IMA J. Appl. Maths 51 (1), 27–67.Google Scholar
De Luca, L., Cardone, G., Aymer de la Chevalerie, D. & Fonteneau, A. 1993 Görtler instability of a hypersonic boundary layer. Exp. Fluids 16, 10–16.Google Scholar
Denier, J. P., Hall, P. & Seddougui, S. O. 1991 On the receptivity problem for Görtler vortices: vortex motions induced by wall roughness. Phil. Trans. R. Soc. Lond. A 335 (1636), 51–85.Google Scholar
El-Hady, N. M. & Verma, A. K. 1983 Growth of Görtler vortices in compressible boundary layers along curved surfaces. J. Engng Appl. Sci. 2 (3), 213–238.Google Scholar
Finnis, M. V. & Brown, A. 1997 The linear growth of Görtler vortices. Intl J. Heat Fluid Flow 18 (4), 389–399.Google Scholar
Flechner, S. G., Jacobs, P. F. & Whitcomb, R. T.1976 A high subsonic speed wind tunnel investigation of winglets on a representative second-generation jet transport wing. NASA TN D-8264.Google Scholar
Floryan, J. M. 1991 On the Görtler instability of boundary layers. Prog. Aerosp. Sci. 28 (3), 235–271.Google Scholar
Floryan, J. M. & Saric, W. S. 1982 Stability of Görtler vortices in boundary layers. AIAA J. 20 (3), 316–324.Google Scholar
Ginoux, J. J. 1971 Streamwise vortices in reattaching high-speed flows: a suggested approach. AIAA J. 9 (4), 759–760.Google Scholar
Goldstein, S. 1938 Modern Developments in Fluid Dynamics: An Account of Theory and Experiment Relating to Boundary Layers, Turbulent Motion and Wakes, vol. 1. Clarendon Press.Google Scholar
Goldstein, M. E. 1978 Unsteady vortical and entropic distortions of potential flows round arbitrary obstacles. J. Fluid Mech. 89, 433–468.Google Scholar
Görtler, H. 1940 Uber eine Dreidimensionale Instabilitat Laminarer Grenzschichten am Konkaven Wanden. Naschr Wiss Gas, Gottingen Math Phys Klasse 2 (1).Google Scholar
Graziosi, P. & Brown, G. L. 2002 Experiments on stability and transition at Mach 3. J. Fluid Mech. 472, 83–124.Google Scholar
Gulyaev, A. N., Kozlov, V. E., Kuzenetsov, V. R., Mineev, B. I. & Sekundov, A. N. 1989 Interaction of a laminar boundary layer with external turbulence. Fluid Dyn. 24 (5), 700–710; Translated from Izv, Akad. Navk. SSSR Mekh. Zhid. Gaza 6.Google Scholar
Hall, P. 1983 The linear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 130, 41–58.Google Scholar
Hall, P. 1990 Görtler vortices in growing boundary layers: the leading edge receptivity problem, linear growth and the nonlinear breakdown stage. Mathematika 37 (74), 151–189.Google Scholar
Hall, P. & Fu, Y. 1989 On the Görtler vortex instability mechanism at hypersonic speeds. Theor. Comput. Fluid Dyn. 1 (3), 125–134.Google Scholar
Hall, P. & Malik, M. 1989 The growth of Görtler vortices in compressible boundary layers. J. Engng Maths 23 (3), 239–251.Google Scholar
Hammerlin, G. 1961 Über die Stabilität einer kompressiblen Strömung längs einer konkaven Wand bei verschiedenen Wand-temperaturverhältnissen. Deutsche Versuchsanstalt für Luftfahrt (176).Google Scholar
Kemp, N.1951 The laminar three-dimensional boundary layer and a study of the flow past a side edge. MSc thesis, Cornell University.Google Scholar
Kobayashi, R. & Kohama, Y. 1977 Taylor–Görtler instability of compressible boundary layers. AIAA J. 15 (12), 1723–1727.Google Scholar
Kottke, V. 1988 On the instability of laminar boundary layers along concave walls towards Görtler vortices. In Propagation in Systems Far from Equilibrium, pp. 390–398. Springer.Google Scholar
Laufer, J. 1954 Factors affecting transition Reynolds numbers on models in supersonic wind tunnels. J. Aero. Sci. 21, 497–498.Google Scholar
Leib, S. J., Wundrow, D. W. & Goldstein, M. E. 1999 Effect of free-stream turbulence and other vortical disturbances on a laminar boundary layer. J. Fluid Mech. 380, 169–203.Google Scholar
Li, F., Choudhari, M., Chang, C.-L., Greene, P. & Wu, M. 2010 Development and breakdown of gortler vortices in high speed boundary layers. In 48th AIAA Aero. Sc. Meeting, p. 705.Google Scholar
Liepmann, H. W.1945 Investigation of boundary layer transition on concave walls. NACA Wartime Rep. W87.Google Scholar
Mangalam, S. M., Dagenhart, J. R., Hepner, T. F. & Meyers, J. F.1985 The Görtler instability on an airfoil. AIAA Paper 85-0491.Google Scholar
Marensi, E. & Ricco, P. 2017 Growth and wall-transpiration control of nonlinear unsteady Görtler vortices forced by free-stream vortical disturbances. Phys. Fluids 29 (11), 114106.Google Scholar
Marensi, E., Ricco, P. & Wu, X. 2017 Nonlinear unsteady streaks engendered by the interaction of free-stream vorticity with a compressible boundary layer. J. Fluid Mech. 817, 80–121.Google Scholar
Matsubara, M. & Alfredsson, P. H. 2001 Disturbance growth in boundary layers subjected to free-stream turbulence. J. Fluid Mech. 430, 149–168.Google Scholar
Mayle, R. E. 1991 The role of laminar-turbulent transition in gas turbine engines. Trans. ASME J. Turbomach. 113 (4), 509–537.Google Scholar
Ovchinnikov, V., Choudhari, M. M. & Piomelli, U. 2008 Numerical simulations of boundary-layer bypass transition due to high-amplitude free-stream turbulence. J. Fluid Mech. 613, 135–169.Google Scholar
Ren, J. & Fu, S. 2015 Secondary instabilities of Görtler vortices in high-speed boundary layer flows. J. Fluid Mech. 781, 388–421.Google Scholar
Ricco, P., Luo, J. & Wu, X. 2011 Evolution and instability of unsteady nonlinear streaks generated by free-stream vortical disturbances. J. Fluid Mech. 677, 1–38.Google Scholar
Ricco, P., Tran, D.-L. & Ye, G. 2009 Wall heat transfer effects on Klebanoff modes and Tollmien–Schlichting waves in a compressible boundary layer. Phys. Fluids 21, 024106.Google Scholar
Ricco, P., Walsh, E. J., Brighenti, F. & McEligot, D. M. 2016 Growth of boundary-layer streaks due to free-stream turbulence. Intl J. Heat Fluid Flow 61, 272–283.Google Scholar
Ricco, P. & Wu, X. 2007 Response of a compressible laminar boundary layer to free-stream vortical disturbances. J. Fluid Mech. 587, 97–138.Google Scholar
Saric, W. S. 1994 Görtler vortices. Annu. Rev. Fluid Mech. 26 (1), 379–409.Google Scholar
Schneider, S. P. 2008 Development of hypersonic quiet tunnels. J. Spacecr. Rockets 45‐4, 641–664.Google Scholar
Spall, R. E. & Malik, M. R. 1989 Görtler vortices in supersonic and hypersonic boundary layers. Phys. Fluids 1 (11), 1822–1835.Google Scholar
Stewartson, K. 1964 The Theory of Laminar Boundary Layers in Compressible Fluids. Claredon Press.Google Scholar
Swearingen, J. D. & Blackwelder, R. F.1983 Parameters controlling the spacing of streamwise vortices on concave walls. AIAA Paper 83-0380.Google Scholar
Tani, I. 1962 Production of longitudinal vortices in the boundary layer along a concave wall. J. Geophys. Res. 67 (8), 3075–3080.Google Scholar
Viaro, S. & Ricco, P. 2018 Neutral stability curves of low-frequency Görtler flow generated by free-stream vortical disturbances. J. Fluid Mech. 845, R1.Google Scholar
Volino, R. J. & Simon, T. W. 1995 Bypass transition in boundary layers including curvature and favorable pressure gradient effects. Trans. ASME J. Turbomach. 117 (1), 166–174.Google Scholar
Wadey, P. D. 1992 On the linear development of Görtler vortices in compressible boundary layers. Eur. J. Mech. (B/Fluids) 11, 705–717.Google Scholar
Wang, Q.-C., Wang, Z.-G. & Zhao, Y.-X. 2018 Visualization of Görtler vortices in supersonic concave boundary layer. J. Vis. 21 (1), 57–62.Google Scholar
Whang, C. W. & Zhong, X.2002 Receptivity of Görtler vortices in hypersonic boundary layers. AIAA Paper 2002-0151.Google Scholar
Whang, C. W. & Zhong, X.2003 Leading edge receptivity of Görtler vortices in a Mach 15 flow over a blunt wedge. AIAA Paper 2003-0790.Google Scholar
Wu, X. & Moin, P. 2009 Direct numerical simulation of turbulence in a nominally zero-pressure-gradient flat-plate boundary layer. J. Fluid Mech. 630, 5–41.Google Scholar
Wu, X., Zhao, D. & Luo, J. 2011 Excitation of steady and unsteady Görtler vortices by free-stream vortical disturbances. J. Fluid Mech. 682, 66–100.Google Scholar
Xu, D., Zhang, Y. & Wu, X. 2017 Nonlinear evolution and secondary instability of steady and unsteady Görtler vortices induced by free-stream vortical disturbances. J. Fluid Mech. 829, 681–730.Google Scholar