Nonlinear and nonstationary modes of the lower branch of the incompressible boundary layer flow due to a rotating disk
Author:
M. Turkyilmazoglu
Journal:
Quart. Appl. Math. 65 (2007), 4368
MSC (2000):
Primary 76U05; Secondary 76E09, 34K25
Published electronically:
February 6, 2007
MathSciNet review:
2313148
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Abstract: In this paper a theoretical study is undertaken to investigate the structure of the lower branch neutral stability modes of threedimensional small disturbances imposed on the incompressible Von Karman's boundary layer flow due to a rotating disk. Particular attention is given to the shortwavelength nonlinear nonstationary crossflow vortex modes at sufficiently high Reynolds numbers with reasonably small scaled frequencies. Following closely the asymptotic frameworks introduced in [Proc. Roy. Soc. London Ser. A 406 (1986), 93106] and [Proc. Roy. Soc. London Ser. A 413 (1987), 497513] for the stationary linear and nonlinear modes, it is revealed here that the nonstationary modes with sufficiently long time scale can also be described by an asymptotic expansion procedure based on the tripledeck theory. Making use of this approach, which takes into account the nonlinear and nonparallel effects, the asymptotic structure of the nonstationary modes is shown to be adjusted by a balance between viscous and Coriolis forces, and resulted from the fact of vanishing shear stress at the disk surface. As a consequence of the matching of the solutions in adjacent regions it is found that in the linear case the wavenumber and the orientation of the lower branch modes are governed by an eigenrelation, which is akin to the one obtained previously in [Proc. Roy. Soc. London Ser. A 406 (1986), 93106] for the stationary modes. The asymptotic theory shows that the nonparallelism has a destabilizing effect. A Landautype equation for the modulated vortex amplitude with coefficients that are often difficult to get from finite Reynolds number computations has also been obtained from a weakly nonlinear analysis in the limit of infinitely large Reynolds numbers. The nonlinearity has also been found to be destabilizing for both positive and negative frequency waves, though finite amplitude growth of a disturbance having positive frequency close to the neutral location is more effective.
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 1.
 P. Hall, An asymptotic investigation of the stationary modes of instability of the boundary layer on a rotatingdisk, Proc. Roy. Soc. London Ser. A, 406 (1986), 93106.
 2.
 S. O. MacKerrel, A nonlinear asymptotic investigation of the stationary modes of instability of the threedimensional boundary layer on a rotating disk, Proc. Roy. Soc. London Ser. A, 413 (1987), 497513. MR 0915128 (89a:76038)
 3.
 R. J. Lingwood, Absolute instability of the boundary layer on a rotating disk, J. Fluid Mech., 299, (1995), 1733. MR 1351381 (96e:76061)
 4.
 R. J. Lingwood, An experimental study of absolute instability of the rotatingdisk boundary layer flow, J. Fluid Mech., 314, (1996), 373405.
 5.
 R. J. Lingwood, On the application of the Briggs' and steepestdescent methods to a boundary layer flow, Stud. Appl. Math., 98, (1997), 213254. MR 1441304 (97k:76034)
 6.
 M. Turkyilmazoglu, Linear absolute and convective instabilities of some two and threedimensional flows, Ph.D. Thesis, (1998), University of Manchester.
 7.
 M. Turkyilmazoglu, J. W. Cole and J. S. B. Gajjar, Absolute and convective instabilities in the compressible boundary layer on a rotating disk, Theoret. Comput. Fluid Dyn., 14, (2000), 2137.
 8.
 M. Turkyilmazoglu and J. S. B. Gajjar, Direct spatial resonance in the laminar boundary layer due to a rotatingdisk, SADHANAACAD P ENG S., 25, (2000), 601617.
 9.
 M. Turkyilmazoglu and J. S. B. Gajjar, An analytic approach for calculating absolutely unstable inviscid modes of the boundary layer on a rotating disk, Stud. Appl. Math., 106, (2001), 419435. MR 1825844 (2002a:76168)
 10.
 B. Pier, Finiteamplitude crossflow vortices, secondary instability and transition in the rotatingdisk boundary layer, J. Fluid Mech., 487, (2003), 315343. MR 2017795 (2004i:76186)
 11.
 A. J. Cooper and P. W. Carpenter, The stability of rotatingdisk boundary layer flow over a compliant wall. Part II. Absolute instability , J. Fluid Mech., 350, (1997), 261270. MR 1481902 (98k:76056)
 12.
 N. Gregory, J. T. Stuart and W. S. Walker, On the stability of threedimensional boundary layers with applications to the flow due to a rotating disk, Philos. Trans. Roy. Soc. London Ser. A, 248, (1955), 155199. MR 0072616 (17:311c)
 13.
 M. R. Malik, S. P. Wilkinson and S. A. Orszag, Instability and transition in rotatingdisk flow, AIAA Journal, 19, (1981), 11311138.
 14.
 M. R. Malik, D. I. A. Poll, Effect of curvature on threedimensional boundarylayer stability, AIAA Journal, 23, (1985), 13621369. MR 0798889 (87a:76058)
 15.
 M. R. Malik, The neutral curve for stationary disturbances in rotatingdisk flow, J. Fluid Mech., 164, (1986), 257287. MR 0844675 (87f:76113)
 16.
 S. P. Wilkinson and M. R. Malik, Stability experiments in rotatingdisk flow, AIAA Pap., 1760, (1983).
 17.
 S. P. Wilkinson and M. R. Malik, Stability experiments in the flow over a rotatingdisk, AIAA Journal, 23, (1985), 588595.
 18.
 A. J. Faller, Instability and transition of disturbed flow over a rotatingdisk, J. Fluid Mech., 230, (1991), 245269.
 19.
 B. I. Federov, G. Z. Plavnik, I. V. Prokhorov and L. G. Zhukhovitskii, Transitional flow conditions on a rotatingdisk, J. Eng. Phys., 31, (1976), 14481453.
 20.
 P. Balakumar and M. R. Malik, Travelling disturbances in rotatingdisk flow, Theoret. Comput. Fluid Dyn., 2, (1990), 125137.
 21.
 K. Stewartson, On the flow near the trailingedge of a flat plate, MATHEMATICA, 16, (1969), 106121.
 22.
 K. Stewartson and P. G. Williams, Selfinduced separation, Proc. Roy. Soc. London Ser. A, 312, (1969), 181206.
 23.
 F. T. Smith, On the high Reynolds number theory of laminar flows, IMA Journal of Applied Mathematics, 28 (1982), 207281. MR 0666155 (83g:76047)
 24.
 Vladimir V. Sychev, A. I. Ruban, Victor V. Sychev and G. L. Korolev, Asymptotic theory of separated flows, Cambridge University Press, (1998). MR 1659235 (2000d:76049)
 25.
 C. C. Lin, On the stability of twodimensional parallel flows. Part III. Stability in a viscous fluid, Quart. J. Mech. Appl. Math., 3, (1946), 277301. MR 0014894 (7:346b)
 26.
 F. T. Smith, On the nonparallel flow stability of the Blasius boundary layer, Proc. Roy. Soc. London Ser. A, 366, (1979), 91109.
 27.
 F. T. Smith and R. J. Bodonyi, Nonlinear critical layers and their development in streamingflow stability, J. Fluid Mech., 118, (1982), 165185. MR 0663988 (83g:76055)
 28.
 A. P. Bassom and J. S. B. Gajjar, Nonstationary crossflow vortices in threedimensional boundarylayer flows, Proc. Roy. Soc. London Ser. A, 417, (1988), 179212. MR 0944282 (89e:76036)
 29.
 M. Turkyilmazoglu and J. S. B. Gajjar, Upper branch nonstationary modes of the boundary layer due to a rotating disk, Applied Mathematics Letters, 14, (2001), 685690. MR 1836070 (2002d:76050)
 30.
 Y. Kohama, Crossflow instability in rotatingdisk boundary layer, AIAA Pap., 1340, (1987).
 31.
 Y. Kohama, Crossflow instability in a spinning disk boundary layer, AIAA Journal, 31, (1992), 212214.
 32.
 S. L. G. Jarre and M. P. Chauve, Experimental study of rotatingdisk instability. I. Natural flow, Phys. Fluids., 8, (1996), 496508.
 33.
 S. L. G. Jarre and M. P. Chauve, Experimental study of rotatingdisk instability. II. Forced flow, Phys. Fluids., 8, (1996), 29852994.
 34.
 J. J. Healey, A new convective instability of the rotatingdisk boundary layer with growth normal to the disk, J. Fluid Mech., 560, (2006), 279310.
 35.
 M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, Dover Publications, (1955).
 36.
 A. I. Ruban, Nonlinear equation for the amplitude of the TollmienSchlichting wave in the boundary layer, Fluid Dyn. Res., 19, (1983), 709716.
 37.
 P. Hall and F. T. Smith, On the effects of nonparallelism, 3dimensionality, and mode interaction in nonlinear boundarylayer stability , Stud. Appl. Math., 71, (1984), 91120.
 38.
 S. N. Timoshin, On shortwavelength instabilities in threedimensional classical boundary layers, Eur. J. Mech., B/Fluids, 14, (1995), 409437. MR 1349727 (96e:76064)
 39.
 M. Turkyilmazoglu, Asymptotic calculation of inviscidly absolutely unstable modes of the compressible boundary layer on a rotating disk, Applied Mathematics Letters, 19, (2006), 795800. MR 2232257
 40.
 J. S. B. Gajjar, Nonlinear evolution of a 1st mode oblique wave in a compressible boundary layer. Part 1. Heated/cooled walls, IMA Journal of Applied Mathematics, 53, (1994), 221248. MR 1314256 (95i:76089)
 41.
 M. Choudhari, Longwavelength asymptotics of stationary crossflow instability in incompressible and compressible boundary layer flows, AIAA Pap., 2137, (1996).
 42.
 S. O. Seddougui, A nonlinear investigation of the stationary modes of instability of the threedimensional compressible boundary layer due to a rotating disk, Quart. J. Mech. Appl. Math., 43, (1990), 467497. MR 1081299 (91j:76055)
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Additional Information
M. Turkyilmazoglu
Affiliation:
Department of Mathematics, Hacettepe University, 06532Beytepe, Ankara, Turkey
Address at time of publication:
Department of Mathematics, Hacettepe University, 06532Beytepe, Ankara, Turkey
Email:
turkyilm@hotmail.com
DOI:
http://dx.doi.org/10.1090/S0033569X0701050X
PII:
S 0033569X(07)01050X
Keywords:
Rotatingdisk flow,
stationary/nonstationary waves,
tripledeck theory
Received by editor(s):
January 27, 2006
Published electronically:
February 6, 2007
Article copyright:
© Copyright 2007 Brown University
The copyright for this article reverts to public domain 28 years after publication.
