Hostname: page-component-6766d58669-76mfw Total loading time: 0 Render date: 2026-05-20T00:54:18.137Z Has data issue: false hasContentIssue false
  • English
  • Français

On unsteady boundary-layer separation in supersonic flow. Part 1. Upstream moving separation point

Published online by Cambridge University Press:  15 April 2011

A. I. RUBAN ,
D. ARAKI ,
R. YAPALPARVI  and
J. S. B. GAJJAR
A. I. RUBAN*
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2BZ, UK
D. ARAKI
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
R. YAPALPARVI
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario L8S 4K1, Canada
J. S. B. GAJJAR
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
*
†Email address for correspondence: a.ruban@imperial.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

This study is concerned with the boundary-layer separation from a rigid body surface in unsteady two-dimensional laminar supersonic flow. The separation is assumed to be provoked by a shock wave impinging upon the boundary layer at a point that moves with speed Vsh along the body surface. The strength of the shock and its speed Vsh are allowed to vary with time t, but not too fast, namely, we assume that the characteristic time scale t ≪ Re−1/2/Vw2. Here Re denotes the Reynolds number, and Vw = −Vsh is wall velocity referred to the gas velocity V∞ in the free stream. We show that under this assumption the flow in the region of interaction between the shock and boundary layer may be treated as quasi-steady if it is considered in the coordinate frame moving with the shock. We start with the flow regime when Vw = O(Re−1/8). In this case, the interaction between the shock and boundary layer is described by classical triple-deck theory. The main modification to the usual triple-deck formulation is that in the moving frame the body surface is no longer stationary; it moves with the speed Vw = −Vsh. The corresponding solutions of the triple-deck equations have been constructed numerically. For this purpose, we use a numerical technique based on finite differencing along the streamwise direction and Chebyshev collocation in the direction normal to the body surface. In the second part of the paper, we assume that 1 ≫ Vw ≫ O(Re−1/8), and concentrate our attention on the self-induced separation of the boundary layer. Assuming, as before, that the Reynolds number, Re, is large, the method of matched asymptotic expansions is used to construct the corresponding solutions of the Navier–Stokes equations in a vicinity of the separation point.

JFM classification

Boundary Layers: Boundary layer separation Aerodynamics: High-speed flow

Information

Type
Papers
Information
Journal of Fluid Mechanics , Volume 678 , 10 July 2011 , pp. 124 - 155
Copyright
Copyright © Cambridge University Press 2011

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.)

Article purchase

Temporarily unavailable

References

REFERENCES

Araki, D. 2006 Boundary-layer separation on a moving surface in supersonic flow. PhD thesis, The University of Manchester.Google Scholar
Battam, N. W., Gorounov, D. G., Korolev, G. L. & Ruban, A. I. 2004 Shock wave interaction with a viscous wake in supersonic flow. J. Fluid Mech. 504, 301–341.CrossRefGoogle Scholar
Blasius, H. 1908 Grenzschichten in flüssigkeiten mit kleiner reibung. Z. Math. Phys. 56 (1), 1–37 (translation in NACA TM 1256).Google Scholar
Cebeci, T. 1979 The laminar boundary layer on a circular cylinder started impulsively from rest. J. Comput Phys. 31, 153–172.CrossRefGoogle Scholar
Cebeci, T. 1982 Unsteady separation. In Numerical and Physical Aspects of Aerodynamic Flows (ed. Cebeci, T.), pp. 265–278. Springer.CrossRefGoogle Scholar
Cebeci, T. 1986 Unsteady boundary layers with an intelligent numerical scheme. J. Fluid Mech. 163, 129–140.CrossRefGoogle Scholar
Collins, W. M. & Dennis, S. C. R. 1973 a Flow past an impulsively started cylinder. J. Fluid Mech. 60, 105–127.CrossRefGoogle Scholar
Collins, W. M. & Dennis, S. C. R. 1973 b The initial flow past an impulsively started circular cylinder. Q. J. Mech. Appl. Maths 26, 53–75.CrossRefGoogle Scholar
Cowley, S. J. 1983 Computer extension and analytic continuation of Blasius' expansion for impulsive flow past a circular cylinder. J. Fluid Mech. 135, 389–405.CrossRefGoogle Scholar
Degani, A. T., Li, Q. & Walker, J. D. A. 1996 Unsteady separation from the leading edge of a thin airfoil. Phys. Fluids 8 (3), 704–714.CrossRefGoogle Scholar
van Dommelen, L. L. & Shen, S. F. 1980 The spontaneous generation of the singularity in a separating laminar boundary layer. J. Comput. Phys. 38 (2), 125–140.CrossRefGoogle Scholar
van Dommelen, L. L. & Shen, S. F. 1982 The genesis of separation. In Numerical and Physical Aspects of Aerodynamic Flows (ed. Cebeci, T.), pp. 293–311. Springer.CrossRefGoogle Scholar
van Dommelen, L. L. & Shen, S. F. 1983 An unsteady interactive separation process. AIAA J. 21 (3), 358–362.CrossRefGoogle Scholar
Elliott, J. W., Smith, F. T. & Cowley, S. J. 1983 Breakdown of boundary layers: (i) on moving surfaces; (ii) in semi-similar unsteady flow; (iii) in fully unsteady flow. Geophys. Astrophys. Fluid Dyn. 25, 77–138.CrossRefGoogle Scholar
Goldstein, S. 1948 On laminar boundary-layer flow near a position of separation. Q. J. Mech. Appl. Maths 1 (1), 43–69.CrossRefGoogle Scholar
Goldstein, S. & Rosenhead, L. N. 1936 Boundary layer growth. Proc. Camb. Phil. Soc. 32, 392–401.CrossRefGoogle Scholar
Hartree, D. R. 1939 A solution of the laminar boundary-layer equation for retarded flow. Aero. Res. Coun. Rep. and Memo. 2426 (issued in 1949).Google Scholar
Howarth, L. 1938 On the solution of the laminar boundary layer equations. Proc. R. Soc. Lond. A 164, 547–579.CrossRefGoogle Scholar
Kaplun, S. 1967 Mechanics and Singular Perturbations: A Collection of Papers. Academic.Google Scholar
Korolev, G. L., Gajjar, J. S. B. & Ruban, A. I. 2002 Once again on the supersonic flow separation near a corner. J. Fluid Mech. 463, 173–199.CrossRefGoogle Scholar
Koromilas, C. A. & Telionis, D. P. 1980 Unsteady laminar separation: an experimental study. J. Fluid Mech. 97 (2), 347–384.CrossRefGoogle Scholar
Landau, L. D. & Lifshitz, E. M. 1944 Mechanics of Continuous Media. Gostekhizdat.Google Scholar
Lighthill, M. J. 1953 On boundary layers and upstream influence. II. Supersonic flows without separation. Proc. R. Soc. Lond. A 217, 478–507.Google Scholar
Ludwig, G. R. 1964 An experimental investigation of laminar separation from a moving wall. AIAA Paper 64-6.CrossRefGoogle Scholar
Messiter, A. F. 1970 Boundary-layer flow near the trailing edge of a flat plate. SIAM J. Appl. Math. 18 (1), 241–257.CrossRefGoogle Scholar
Moore, F. K. 1958 On the separation of the unsteady laminar boundary layer. In Boundary Layer Research (ed. Görtler, H.), pp. 296–311. Springer.Google Scholar
Nakayama, Y. 1988 Visualized Flow. Fluid Motion in Basic and Engineering Situations Revealed by Flow Visualization. Compiled by the Japan Society of Mechanical Engineers. Pergamon Press.Google Scholar
Neiland, V. Y. 1969 Theory of laminar boundary layer separation in supersonic flow. Izv. Akad. Nauk SSSR, Mech. Zhidk. Gaza 53–57.Google Scholar
Peridier, V. J., Smith, F. T. & Walker, J. D. A. 1991 Vortex-induced boundary-layer separation. Part 1. The unsteady limit problem Re → ∞. J. Fluid Mech. 232, 99–131.CrossRefGoogle Scholar
Prandtl, L. 1904 Über flüssigkeitsbewegung bei sehr kleiner Reibung. In Verh. III. Intern. Math. Kongr., Heidelberg, pp. 484–491. Teubner, 1905.Google Scholar
Proudman, I. & Johnson, K. 1962 Boundary-layer growth near a rear stagnation point. J. Fluid Mech. 12, 161–168.CrossRefGoogle Scholar
Puhak, R. I., Degani, A. T. & Walker, J. D. A. 1995 Unsteady separation and heat transfer upstream of obstacles. J. Fluid Mech. 305, 1–27.CrossRefGoogle Scholar
Robins, A. J. & Howarth, J. A. 1972 Boundary-layer development at a two-dimensional rear stagnation point. J. Fluid Mech. 56, 161–171.CrossRefGoogle Scholar
Rott, N. 1956 Unsteady viscous flow in the vicinity of a stagnation point. Q. J. Appl. Math. 13, 444–451.CrossRefGoogle Scholar
Sears, W. R. 1956 Some recent developments in airfoil theory. J. Aeronaut. Sci. 23 (5), 490–499.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. & 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
Sychev, V. V. 1972 Laminar separation. Mekh. Zhid. Gaza, No. 3, 47–59 (translation in Fluid Dyn. 7 (3), 407–417).Google Scholar
Sychev, V. V. 1978 Breakdown of plane laminar wake. Uch. Zap. TsAGI 9 (6), 9–16.Google Scholar
Sychev, V. V. 1979 Asymptotic theory of nonstationary separation. Mekh. Zhid. Gaza, No. 6, 21–32 (translation in Fluid Dyn. 14 (6), 829–838).Google Scholar
Sychev, V. V. 1980 On certain singularities in solutions of the boundary-layer equations on a moving surface. Prikl. Mat. Mekh. 44 (5), 831–838 (translation in J. Appl. Math. Mech. 44 (5), 587–591).Google Scholar
Sychev, V. V. 1984 On the asymptotic theory of laminar separation from a moving surface. Prikl. Mat. Mekh. 48 (2), 247–253 (translation in J. Appl. Math. Mech. 48 (2), 171–176).Google Scholar
Sychev, V. V. 1987 Analytical solution of the problem of flow near the boundary-layer separation point on a moving wall. Prikl. Mat. Mekh. 51 (3), 519–521 (translation in J. Appl. Math. Mech. 51 (3), 405–407).Google Scholar
Telionis, D. P. & Tsahalis, D. T. 1974 Unsteady laminar separation over impulsively moved cylinders. Acta Astronautica 1, 1487–1505.CrossRefGoogle Scholar
Telionis, D. P. & Werle, M. J. 1973 Boundary layer separation from moving boundaries. J. Appl. Mech. 95, 369–374.CrossRefGoogle Scholar
Wang, K. C. 1979 Unsteady boundary layer separation. Tech. Rep. MML TR 79-16C. Martin Marietta Laboratory, Baltimore, Maryland, USA.Google Scholar
Wang, K. C. 1982 On the current controversy about unsteady separation. In Numerical and Physical Aspects of Aerodynamic Flows (ed. Cebeci, T.), pp. 279–291. Springer.CrossRefGoogle Scholar
Wundt, H. 1955 Wachstum der laminaren Grenzchicht an schräg angeströmten Zylinder bei Anfahrt aus der Ruhe. Ing.-Arch 23, 212–230.CrossRefGoogle Scholar