Non-linear and non-stationary modes of the lower branch of the incompressible boundary layer flow due to a rotating disk

Turkyilmazoglu, M.

doi:10.1090/S0033-569X-07-01050-X

Author: M. Turkyilmazoglu
Journal: Quart. Appl. Math. 65 (2007), 43-68
MSC (2000): Primary 76U05; Secondary 76E09, 34K25
DOI: https://doi.org/10.1090/S0033-569X-07-01050-X
Published electronically: February 6, 2007
MathSciNet review: 2313148
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper a theoretical study is undertaken to investigate the structure of the lower branch neutral stability modes of three-dimensional small disturbances imposed on the incompressible Von Karman’s boundary layer flow due to a rotating disk. Particular attention is given to the short-wavelength non-linear non-stationary cross-flow 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), 93–106] and [Proc. Roy. Soc. London Ser. A 413 (1987), 497–513] for the stationary linear and non-linear modes, it is revealed here that the non-stationary modes with sufficiently long time scale can also be described by an asymptotic expansion procedure based on the triple-deck theory. Making use of this approach, which takes into account the non-linear and non-parallel effects, the asymptotic structure of the non-stationary 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), 93–106] for the stationary modes. The asymptotic theory shows that the non-parallelism has a destabilizing effect. A Landau-type 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 non-linear analysis in the limit of infinitely large Reynolds numbers. The non-linearity 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.

References

P. Hall, An asymptotic investigation of the stationary modes of instability of the boundary layer on a rotating-disk, Proc. Roy. Soc. London Ser. A, 406 (1986), 93–106.

Sharon MacKerrell, A nonlinear, asymptotic investigation of the stationary modes of instability of the three-dimensional boundary layer on a rotating disc, Proc. Roy. Soc. London Ser. A 413 (1987), no. 1845, 497–513. MR 915128

R. J. Lingwood, Absolute instability of the boundary layer on a rotating disk, J. Fluid Mech. 299 (1995), 17–33. MR 1351381, DOI https://doi.org/10.1017/S0022112095003405

R. J. Lingwood, An experimental study of absolute instability of the rotating-disk boundary layer flow, J. Fluid Mech., 314, (1996), 373–405.

R. J. Lingwood, On the application of the Briggs’ and steepest-descent methods to a boundary-layer flow, Stud. Appl. Math. 98 (1997), no. 3, 213–254. MR 1441304, DOI https://doi.org/10.1111/1467-9590.00048

M. Turkyilmazoglu, Linear absolute and convective instabilities of some two- and three-dimensional flows, Ph.D. Thesis, (1998), University of Manchester.

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), 21–37.

M. Turkyilmazoglu and J. S. B. Gajjar, Direct spatial resonance in the laminar boundary layer due to a rotating-disk, SADHANA-ACAD P ENG S., 25, (2000), 601–617.

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), no. 4, 419–435. MR 1825844, DOI https://doi.org/10.1111/1467-9590.00173

Benoît Pier, Finite-amplitude crossflow vortices, secondary instability and transition in the rotating-disk boundary layer, J. Fluid Mech. 487 (2003), 315–343. MR 2017795, DOI https://doi.org/10.1017/S0022112003004981

A. J. Cooper and Peter W. Carpenter, The stability of rotating-disc boundary-layer flow over a compliant wall. II. Absolute instability, J. Fluid Mech. 350 (1997), 261–270. MR 1481902, DOI https://doi.org/10.1017/S0022112097006964

N. Gregory, J. T. Stuart, and W. S. Walker, On the stability of three-dimensional boundary layers with application to the flow due to a rotating disk, Philos. Trans. Roy. Soc. London Ser. A 248 (1955), 155–199. MR 72616, DOI https://doi.org/10.1098/rsta.1955.0013

M. R. Malik, S. P. Wilkinson and S. A. Orszag, Instability and transition in rotating-disk flow, AIAA Journal, 19, (1981), 1131–1138.

M. R. Malik and D. I. A. Poll, Effect of curvature on three-dimensional boundary-layer stability, AIAA J. 23 (1985), no. 9, 1362–1369. MR 798889, DOI https://doi.org/10.2514/3.9093

Mujeeb R. Malik, The neutral curve for stationary disturbances in rotating-disk flow, J. Fluid Mech. 164 (1986), 257–287. MR 844675, DOI https://doi.org/10.1017/S0022112086002550

S. P. Wilkinson and M. R. Malik, Stability experiments in rotating-disk flow, AIAA Pap., 1760, (1983).

S. P. Wilkinson and M. R. Malik, Stability experiments in the flow over a rotating-disk, AIAA Journal, 23, (1985), 588–595.

A. J. Faller, Instability and transition of disturbed flow over a rotating-disk, J. Fluid Mech., 230, (1991), 245–269.

B. I. Federov, G. Z. Plavnik, I. V. Prokhorov and L. G. Zhukhovitskii, Transitional flow conditions on a rotating-disk, J. Eng. Phys., 31, (1976), 1448–1453.

P. Balakumar and M. R. Malik, Travelling disturbances in rotating-disk flow, Theoret. Comput. Fluid Dyn., 2, (1990), 125–137.

K. Stewartson, On the flow near the trailing-edge of a flat plate, MATHEMATICA, 16, (1969), 106–121.

K. Stewartson and P. G. Williams, Self-induced separation, Proc. Roy. Soc. London Ser. A, 312, (1969), 181–206.

F. T. Smith, On the high Reynolds number theory of laminar flows, IMA J. Appl. Math. 28 (1982), no. 3, 207–281. MR 666155, DOI https://doi.org/10.1093/imamat/28.3.207

Vladimir V. Sychev, Anatoly I. Ruban, Victor V. Sychev, and Georgi L. Korolev, Asymptotic theory of separated flows, Cambridge University Press, Cambridge, 1998. Translated from the 1987 Russian original by Elena V. Maroko and revised by the authors. MR 1659235

C. C. Lin, On the stability of two-dimensional parallel flows. III. Stability in a viscous fluid, Quart. Appl. Math. 3 (1946), 277–301. MR 14894, DOI https://doi.org/10.1090/S0033-569X-1946-14894-3

F. T. Smith, On the non-parallel flow stability of the Blasius boundary layer, Proc. Roy. Soc. London Ser. A, 366, (1979), 91–109.

F. T. Smith and R. J. Bodonyi, Nonlinear critical layers and their development in streaming-flow stability, J. Fluid Mech. 118 (1982), 165–185. MR 663988, DOI https://doi.org/10.1017/S0022112082001013

A. P. Bassom and J. S. B. Gajjar, Nonstationary cross-flow vortices in three-dimensional boundary-layer flows, Proc. Roy. Soc. London Ser. A 417 (1988), no. 1852, 179–212. MR 944282

M. Turkyilmazoglu and J. S. B. Gajjar, Upper branch nonstationary modes of the boundary layer due to a rotating disk, Appl. Math. Lett. 14 (2001), no. 6, 685–690. MR 1836070, DOI https://doi.org/10.1016/S0893-9659%2801%2980027-6

Y. Kohama, Crossflow instability in rotating-disk boundary layer, AIAA Pap., 1340, (1987).

Y. Kohama, Crossflow instability in a spinning disk boundary layer, AIAA Journal, 31, (1992), 212–214.

S. L. G. Jarre and M. P. Chauve, Experimental study of rotating-disk instability. I. Natural flow, Phys. Fluids., 8, (1996), 496–508.

S. L. G. Jarre and M. P. Chauve, Experimental study of rotating-disk instability. II. Forced flow, Phys. Fluids., 8, (1996), 2985–2994.

J. J. Healey, A new convective instability of the rotating-disk boundary layer with growth normal to the disk, J. Fluid Mech., 560, (2006), 279–310.

M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, Dover Publications, (1955).

A. I. Ruban, Nonlinear equation for the amplitude of the Tollmien-Schlichting wave in the boundary layer, Fluid Dyn. Res., 19, (1983), 709–716.

P. Hall and F. T. Smith, On the effects of nonparallelism, 3-dimensionality, and mode interaction in nonlinear boundary-layer stability , Stud. Appl. Math., 71, (1984), 91–120.

S. N. Timoshin, On short-wavelength instabilities in three-dimensional classical boundary layers, European J. Mech. B Fluids 14 (1995), no. 4, 409–437. MR 1349727

M. Turkyilmazoglu, Asymptotic calculation of inviscidly absolutely unstable modes of the compressible boundary layer on a rotating disk, Appl. Math. Lett. 19 (2006), no. 8, 795–800. MR 2232257, DOI https://doi.org/10.1016/j.aml.2005.10.008

J. S. B. Gajjar, Nonlinear evolution of a first mode oblique wave in a compressible boundary layer. I. Heated/cooled walls, IMA J. Appl. Math. 53 (1994), no. 3, 221–248. MR 1314256, DOI https://doi.org/10.1093/imamat/53.3.221

M. Choudhari, Long-wavelength asymptotics of stationary crossflow instability in incompressible and compressible boundary layer flows, AIAA Pap., 2137, (1996).

Sharon O. Seddougui, A nonlinear investigation of the stationary modes of instability of the three-dimensional compressible boundary layer due to a rotating disc, Quart. J. Mech. Appl. Math. 43 (1990), no. 4, 467–497. MR 1081299, DOI https://doi.org/10.1093/qjmam/43.4.467