Influence of suction/blowing on the finite amplitude disturbances having non-stationary modes of a compressible boundary layer flow
Author:
M. Turkyilmazoglu
Journal:
Quart. Appl. Math. 66 (2008), 447-471
MSC (2000):
Primary 76U05; Secondary 76E09, 34K25
DOI:
https://doi.org/10.1090/S0033-569X-08-01093-3
Published electronically:
June 4, 2008
MathSciNet review:
2445523
Full-text PDF Free Access
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Abstract: In this paper, a weakly non-linear stability analysis is pursued to explore the effects of suction and blowing on a compressible mode of instability of the three-dimensional boundary layer flow induced by a rotating disk. The main thrust of the research is to extend the stationary work of Seddougui and Bassom (1996) to cover for the non-stationary mode so that it is targeted to determine whether the major role in finite amplitude destabilization of the boundary layer is played by the stationary mode of Seddougui and Bassom (1996) or the non-stationary mode as calculated from the present study. Within this perspective, the basic compressible flow obtained in the large Reynolds number limit, together with a suction parameter entering into the wall with normal velocity at the wall, is perturbed by disturbances which are constituted as the non-linear interaction of fundamental modes and harmonics. The effects of non-linearity are then explored by deriving a finite amplitude equation governing the evolution of the non-linear lower branch modes and also allowing the finite amplitude growth of a disturbance close to the neutral location determined from a prior stability analysis. Although the form of the amplitude equation is not at all surprising (given a similar result for the stationary vortices in Seddougui and Bassom (1996)), the dependence of the coefficients of the Landau-type modulated vortex amplitude equation on the frequency parameter is established here. A close examination of the coefficients of this evolution equation indicates strongly that the non-linearity has a destabilizing effect for both suction and blowing through the surface of the disk, the effect of which is much more stronger for a suction. Also the impact of non-linearity is higher for the non-stationary compressible modes than for the stationary waves of Seddougui (1990) and Seddougui and Bassom (1996). Moreover, in the case of suction, there occurs a regime of non-stationary modes that cover not only the positive frequency waves, but also waves having negative frequencies, and these modes are always unstable no matter whether the wall is insulated or isothermal. The solution of the asymptotic amplitude equation further demonstrates that as the local Mach number increases, compressibility has the influence of stabilization by requiring smaller initial amplitude of the disturbance for the laminar rotating disk boundary layer flow to become unstable whenever fluid injection is applied. Unlike this, suction makes the underlying flow more convectively unstable as far as the compressibility is concerned, particulary for the modes generated as a consequence of an isothermal wall. Both for the suction and injection cases, disturbances having positive frequency are always shown to cause an instantaneous non-linear amplification prior to the negative frequency waves.
References
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- S. P. Wilkinson and M. R. Malik, Stability experiments in rotating-disk flow, AIAA Pap., 1760 (1983).
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- R. Kobayashi, Y. Kohama and Ch. Takamadate, Spiral vortices in boundary layer transition regime on a rotating-disk, Acta Mech., 35 (1980), 71–82.
- A. J. Faller, Instability and transition of disturbed flow over a rotating-disk, J. Fluid Mech., 230 (1991), 245–269.
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- K. Stewartson and P. G. Williams, Self-induced separation, Proc. Roy. Soc. London Ser. A, 312 (1969), 181–206.
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- S. L. G. Jarre and M. P. Chauve, Experimental study of rotating-disk instability. II. Forced flow, Phys. Fluids, 8 (1996), 2985–2994.
- M. Turkyilmazoglu, Lower branch modes of the compressible boundary layer flow due to a rotating-disk, Stud. Appl. Maths., 114 (2005), 17–43. MR 2117326 (2005h:76033)
- M. Turkyilmazoglu, Non-linear and non-stationary modes of the lower branch of the incompressible boundary layer flow due to a rotating-disk, Quart. Appl. Math., 65 (2007), 43–68. MR 2313148
- K. Stewartson, The theory of laminar boundary layers in compressible fluids, Oxford University Press (1964).
- N. Riley, The heat transfer from a rotating-disk, Q. J. Mech. Appl. Math., 17 (1964), 331–349. MR 0169488 (29:6736)
- M. Turkyilmazoglu and N. Uygun, Compressible modes of the rotating disk boundary layer flow leading to absolute instability, Stud. Appl. Maths., 115 (2005), 1–20. MR 2141274 (2005m:76153)
- M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, Dover Publications (1955).
- M. Turkyilmazoglu, Effects of mass transfer on the non-stationary lower branch modes of a compressible boundary layer flow, Stud. Appl. Maths., 115 (2005), 357–385. MR 2172694 (2006e:76100)
- M. R. Dhanak, Effects of uniform suction on the stability of flow on a rotating disc, Proc. Roy. Soc. London Ser. A, 439 (1992), 431.
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Additional Information
M. Turkyilmazoglu
Affiliation:
Department of Mathematics, Hacettepe University, 06532-Beytepe, Ankara, Turkey
Email:
turkyilm@hotmail.com
Keywords:
Rotating-disk flow,
suction/blowing,
linear/non-linear waves,
compressible boundary layer
Received by editor(s):
January 22, 2007
Published electronically:
June 4, 2008
Article copyright:
© Copyright 2008
Brown University