Abstract
A spectral collocation method is used to obtain the solution to the Orr-Sommerfeld stability equation. The accuracy of the method is established by comparing against well documented flows, such as the plane Poiseuille and the Blasius Boundary layers.
The focus is then placed on the generalised Hiemenz flow, an exact solution to the Navier-Stokes equations constituting the base flow at the leading edge of swept cylinders and aerofoils. The spanwise profile of this flow is very similar to that of Blasius but, unlike the latter case, there is no rational approximation leading to the Orr-Sommerfeld equation.
We will show that if, based on experimentally obtained intuition, a nonrational reduction of the full system of linear stability equations is attempted and the resulting Orr-Sommerfeld equation is solved, the linear stability critical Reynolds number is overestimated, as has indeed been done in the past.
However, as shown by recent Direct Numerical Simulation results, the frequency eigenspectrum of instability waves may still be obtained through solution of the Orr-Sommerfeld equation. This fact lends some credibility to the assumption under which the Orr-Sommerfeld equation is obtained insofar as the identification of the frequency regime responsible for linear growth is concerned.
Finally, an argument is presented pointing towards potential directions in the ongoing research for explanation of subcriticality in the leading edge boundary layer.
Similar content being viewed by others
References
K. Hiemenz, Dingl.Polytechn. J. 326 (1911) 321.
H. Schlichting, Boundary Layer Theory. McGraw-Hill (1979).
D.I.A. Poll, College of Aeronautics Rep. 7805 (1978).
D.I.A. Poll, The Aeronautical Quart 30 (1979) 607.
D.I.A. Poll, I.U.T.A.M. Symposium on Laminar-Turbulent Transition. Stuttgart, Springer Verlag (1980).
D.I.A. Poll, The Aeronautical Quart. 34 (1983) 1.
P. Hall, M.R. Malik and D.I.A. Poll, NASA CR-172504 (1984).
P. Hall and M.R. Malik, J. Fluid Mech. 163 (1986) 257.
P.R. Spalart, AGARD CP-438 (1988) 1–5.
J. Jiménez, C. Martel, J.C. Agüi and J.A. Zufiria, ETSIA MF-903 (1990).
V. Theofilis, Ph.D. Thesis, Dept. of Aeronautical Engineering, University of Manchester (1991).
V. Theofilis, Int. J. Numer. Methods Fluids 16 (1993) 153.
C.C. Lin, The Theory of Hydrodynamic Stability, CUP (1955).
S.A. Orszag, J. Fluid Mech. 50 (1971) 689.
M.G. Macaraeg, C.L. Street and M.Y. Hussaini, NASA TP-2858 (1988).
M.R. Malik, NASA-CR-165952 (1982).
Th.L. van Stijn and A.I. van de Vooren, J. Eng. math. 14 (1980) 17.
D. Gottlieb, M.Y. Hussaini and S.A. Orszag, in Spectral Methods for Partial Differential Equations. (eds. Voigt, R.G. Gottlieb, D. and Hussaini, M.Y.), SIAM Philadelphia, p. 1 (1984).
T.A. Zang, CFD Lecture Series 89–04. Von Karman Institute for Fluid Dynamics, Brussels (1989).
J.P. Boyd, Chebyshev and Fourier Spectral Methods, Lecture Notes in Engineering 49, Springer (1989).
J.H. Wilkinson, The algebraic Eigenvalue Problem. Clarendon (1965).
Numerical Algorithms Group, Mark 15 (1992).
C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods in Fluid Dynamics. Springer (1988).
L.M. Mack, J. Fluid Mech. 73 (1976) 497.
C.L. Street, T.A. Zang and M.Y. Hussaini, AIAA Pap. No. 84-0170 (1984).
V. Theofilis, Master Class Lecture Notes. Mathematics Research Institute, the Netherlands (1993).
L. Rosenhead, Laminar Boundary Layers. Oxford University Press (1963).
P.W. Duck, J. Fluid Mech. 160 (1985) 465.
P.W. Duck and O.R. Burggraf, J. Fluid Mech. 162 (1986) 1.
O.R. Burggraf and P.W. Duck, in Numerical and Physical Aspects of Aerodynamical Flows. (T. Cebeci ed.), Springer Verlag NY. (1981) 195.
S.C.R. Dennis, D.B. Ingham and R.N. Cook, J. Comp. Phys. 33 (1979) 325.
V. Theofilis, On subcritical instability of the attachment line boundary layer. Submitted to J. Fluid Mech.
D. Arnal, AGARD Rep. 793, Von Karman Institute for Fluid Dynamics. Brussels (1993).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Theofilis, V. The discrete temporal eigenvalue spectrum of the generalised Hiemenz flow as solution of the Orr-Sommerfeld equation. J Eng Math 28, 241–259 (1994). https://doi.org/10.1007/BF00058439
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00058439