Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-17T14:16:37.699Z Has data issue: false hasContentIssue false
  • English
  • Français

On receptivity of marginally separated flows

Published online by Cambridge University Press:  17 November 2020

K. Jain Open the ORCID record for K. Jain [Opens in a new window] ,
A. I. Ruban Open the ORCID record for A. I. Ruban [Opens in a new window]  and
S. Braun Open the ORCID record for S. Braun [Opens in a new window]
K. Jain*
Affiliation:
Center for Cybersecurity Systems and Networks, Amrita Vishwa Vidyapeetham, Amritapuri690525, India
A. I. Ruban
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen's Gate, LondonSW7 2BZ, UK
S. Braun
Affiliation:
Institute of Fluid Mechanics and Heat Transfer, TU Wien, Getreidemarkt 9, 1060Wien, Austria
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we study the receptivity of the boundary layer to suction/blowing in marginally separated flows, like the one on the leading edge of a thin aerofoil. We assume that the unperturbed laminar flow is two-dimensional, and investigate the response of the boundary layer to two-dimensional as well as to three-dimensional perturbations. In both cases, the perturbations are assumed to be weak and periodic in time. Unlike conventional boundary layers, the marginally separated boundary layers cannot be treated using the quasi-parallel approximation. This precludes the normal-mode representation of the perturbations. Instead, we had to solve the linearised integro-differential equation of the marginal separation theory, which was done numerically. For two-dimensional perturbations, the results of the calculations show that the perturbations first grow in the inside of the separation region, but then start to decay downstream. For three-dimensional perturbations, instead of dealing with the integro-differential equation of marginal separation, we found it convenient to work with the Fourier transforms of the fluid-dynamic functions. The equations for the Fourier transforms are also solved numerically. Our calculations show that a three-dimensional wave packet forms downstream of the source of perturbations in the boundary layer.

JFM classification

Boundary Layers: Boundary layer receptivity Boundary Layers: Boundary layer separation
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 907 , 25 January 2021 , A1
Check for updates
Copyright
© The Author(s), 2020. Published by 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

REFERENCES

Braun, S. & Kluwick, A. 2004 Unsteady three-dimensional marginal separation caused by surface mounted obstacles and/or local suction. J. Fluid Mech. 514, 121–152.CrossRefGoogle Scholar
Braun, S. & Scheichl, S. 2014 On recent developments in marginal separation theory. Phil. Trans. R. Soc. Lond. A 372, 20130343.Google ScholarPubMed
Brown, S. N. 1985 Marginal separation of a three-dimensional boundary layer on a line of symmetry. J. Fluid Mech. 158, 95–111.CrossRefGoogle Scholar
Duck, P. W., Ruban, A. I. & Zhikharev, C. N. 1996 The generation of Tollmien–Schlichting waves by free stream turbulence. J. Fluid Mech. 312, 341–371.CrossRefGoogle Scholar
Ely, W. L. & Herring, R. N. 1978 Laminar leading edge stall prediction for thin airfoils. AIAA Paper 78-1222. American Institute of Aeronautics and Astronautics.CrossRefGoogle Scholar
Fomina, I. G. 1983 On asymptotic flow theory near corner points on a solid surface. Uch. Zap. TsAGI 14 (5), 31–38.Google Scholar
Goldstein, M. E. 1985 Scattering of acoustic waves into Tollmien–Schlichting waves by small streamwise variations in surface geometry. J. Fluid Mech. 154, 509–529.CrossRefGoogle Scholar
Lin, C. C. 1946 On the stability of two-dimensional parallel flows. Pt. 3. Stability in a viscous fluid. Q. Appl. Maths 277 (3), 117–142.Google Scholar
Messiter, A. F. 1970 Boundary-layer flow near the trailing edge of a flat plate. SIAM J. Appl. Maths 18 (1), 241–257.CrossRefGoogle Scholar
Negoda, V. V. & Sychev, V. V. 1986 The boundary layer on a rapidly rotating cylinder. Izv. Akad. Nauk SSSR Mech. Zhidk. Gaza (5), 36–45.Google Scholar
Neiland, V. Ya. 1969 Theory of laminar boundary layer separation in supersonic flow. Izv. Akad. Nauk SSSR Mech. Zhidk. Gaza (4), 53–57.Google Scholar
Prandtl, L. 1904 Über flüssigkeitsbewegung bei sehr kleiner Reibung. In Verh. III. Intern. Math. Kongr., Heidelberg, pp. 484–491. Teubner, 1905.Google Scholar
Ruban, A. I. 1981 Singular solution of boundary layer equations which can be extended continuously through the point of zero surface friction. Izv. Akad. Nauk SSSR Mech. Zhidk. Gaza (6), 42–52.Google Scholar
Ruban, A. I. 1982 a Asymptotic theory of short separation regions at the leading edge of a thin aerofoil. Izv. Akad. Nauk SSSR Mech. Zhidk. Gaza (1), 42–51.Google Scholar
Ruban, A. I. 1982 b Stability of the preseparation boundary layer at the leading edge of a thin airfoil. Izv. Akad. Nauk SSSR Mech. Zhidk. Gaza (6), 55–62.Google Scholar
Ruban, A. I. 1984 On the generation of Tollmien–Schlichting waves by sound. Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza (5), 44–52.Google Scholar
Ruban, A. I. 2018 Fluid Dynamics. Part 3. Boundary Layers. Oxford University Press.Google Scholar
Ruban, A. I., Bernots, T. & Kravtsova, M. A. 2016 Linear and nonlinear receptivity of the boundary layer in transonic flows. J. Fluid Mech. 768, 154–189.CrossRefGoogle Scholar
Ruban, A. I., Bernots, T. & Pryce, D. 2013 Receptivity of the boundary layer to vibrations of the wing surface. J. Fluid Mech. 723, 480–528.CrossRefGoogle Scholar
Ryzhov, O. S. & Smith, F. T. 1984 Short-length instabilities, breakdown and initial value problems in dynamic stall. Mathematika 31 (2), 163–177.CrossRefGoogle Scholar
Scheichl, S., Braun, S. & Kluwick, A. 2008 On a similarity solution in the theory of unsteady marginal separation. Acta Mechanica 201, 153–170.CrossRefGoogle Scholar
Schubauer, G. B. & Skramsted, H. K. 1948 Laminar-boundary-layer oscillations and transition on a flat plate. NACA Tech. Rep. 909.Google Scholar
Smith, F. T. 1979 On the non-parallel flow stability of the blasius boundary layer. Proc. R. Soc. Lond. A 366, 91–109.Google Scholar
Smith, F. T. 1982 Concerning dynamic stall. Aeronaut. Q. 33 (4), 331–352.CrossRefGoogle Scholar
Stewartson, K. 1969 On the flow near the trailing edge of a flat plate. Mathematika 16 (1), 106–121.CrossRefGoogle Scholar
Stewartson, K., Smith, F. T. & Kaups, K. 1982 Marginal separation. Stud. Appl. Maths 67 (1), 45–61.CrossRefGoogle Scholar
Stewartson, K. & Williams, P. G. 1969 Self-induced separation. Proc. R. Soc. Lond. A 312, 181–206.Google Scholar
Sychev, V. V., Ruban, A. I., Sychev, V. V. & Korolev, G. L. 1998 Asymptotic Theory of Separated Flows. Cambridge University Press.CrossRefGoogle Scholar
Tani, I. 1964 Low-speed flows involving bubble separations. Prog. Aeronaut. Sci. 5, 70–103.CrossRefGoogle Scholar
Terent'ev, E. D. 1981 The linear problem of a vibrator in a subsonic boundary layer. Prikl. Mat. Mekh. 45 (6), 1049–1055.Google Scholar
Ward, J. W. 1963 The behaviour and effects of laminar separation bubbles on aerofoils in incompressible flow. J. R. Aero. Soc. 67, 783–790.CrossRefGoogle Scholar
Wu, X. 2001 Receptivity of boundary layers with distributed roughness to vortical and acoustic disturbances; a second-order asymptotic theory and comparison with experiments. J. Fluid Mech. 431, 91–133.CrossRefGoogle Scholar
Zametaev, V. B. 1986 Existence and non-uniqueness of local separation region in viscous jets. Izv. Akad. Nauk SSSR Mech. Zhidk. Gaza (1), 38–45.Google Scholar