Convective and Absolute Instabilities in the Incompressible Boundary Layer on a Rotating-Disk

Year 2006, Volume: 35 Issue: 2, 117 - 146, 01.02.2006

M. Türkyilmazoglu

Abstract

Convective and Absolute Instabilities in the Incompressible Boundary Layer on a Rotating-Disk

Abstract

The linear absolute/convective instability mechanisms of the incompressible Von Karman‘s boundary layer flow over a rotating-disk are revisited in the present paper in order to review and assemble the available results in the literature on the topic. For this purpose the linearized system of stability equations of motion is first treated numerically, by employing a Spectral method based on Chebyshev collocation as well as a fourth-order Runge-Kutta method in combination with a shooting strategy. Inviscid/viscous stationary and travelling modes which lead to both convective and absolute instability mechanisms were successfully reproduced and compare favorably with those obtained by previous investigators. The validation of the zero-frequency upper-branch modes was also accomplished by the asymptotic expansion technique of [17], which is later extended to cover the non-zero frequency disturbances. The importance of the present study lies in understanding the roles of possible instability mechanisms on the laminar-turbulent transition phenomenon in the three-dimensional boundary layer flow over a rotating-disk, as well as related aerodynamic bodies.

Keywords

Rotating-Disk Flow, Absolute Versus Convective Instability, Matched Asymptotic Expansion, Laminar-Turbulent Transition, 2000 AMS Classification: 76E15

References

• Balakumar, P. and Malik, M. R. Travelling disturbances in rotating-disk flow, Theoret. Com- put. Fluid Dyn. 2, 125–137, 1990.
• Bassom, A. P. and Gajjar, J. S. B. Non-stationary cross-flow vortices in a three-dimensional boundary layer, Proc. Roy. Soc. London Ser. A 417, 179–212, 1988.
• Benney, D. J. and Gustavsson, L. H. A new mechanism for linear and non-linear hydrody- namic instability, Stud. Appl. Math. 64, 185–209, 1981.
• Bers, A. Linear waves and instabilities, Physique des Plasmas., 117–225, 1975.
• Betchov, R. and Criminale, W. O. Spatial instability of the inviscid jet and wake, Phys. Fluids. 9, 359–362, 1966.
• Briggs, R. J. Electron-Stream Interaction With Plasmas (MIT Press, 1964).
• Cole, J. W. Hdrodynamic stability of compressible flows, PhD thesis (University of Exeter, 1995).
• Cooper, A. J. and Carpenter, P. W. The stability of rotating-disk boundary-layer flow over a compliant wall. part i. type i and ii instabilities, J. Fluid Mech. 350, 231–259, 1997.
• Cooper, A. J. and Carpenter, P. W. The stability of rotating-disk boundary-layer flow over a compliant wall. part ii. absolute instability, J. Fluid Mech. 350, 261–270, 1997.
• Davies, C. and Carpenter, P. W. A novel velocity-vorticity formulation of the navier-stokes equations with application to boundary layer disturbance evolution, J. Comput. Phys. 172, 119–165, 2001. [11] Davies, C. and Carpenter, P. W. Global behaviour corresponding to the absolute instability of the rotating-disc boundary layer, J. Fluid Mech. 486, 287–329, 2003.
• Faller, A. J. and Kaylor, R. E. A numerical study of the instability of the laminar ekman boundary layer, J. Atm. Sci. 23, 466–480, 1966.
• Federov, B. I., Plavnik, G. Z., Prokhorov, I. V. and Zhukhovitskii, L. G. Transitional flow conditions on a rotating-disk, J. Eng. Phys. 31, 1448–1453, 1976.
• Gajjar, J. S. B. Nonlinear evolution of a 1st mode oblique wave in a compressible boundary- layer. Part 1. Heated cooled walls, IMA Journal of Applied Mathematics 53, 221–248, 1994. [15] Jasmine, H. A. and Gajjar, J. S. B. Convective and absolute instability in the incompressible boundary layer on a rotating disk in the presence of a uniform magnetic field, Journal of Engineering Mathematics 52, 337–353, 2005.
• Gregory, N., Stuart, J. T. and Walker, W. S. On the stability of three-dimensional boundary layers with applications to the flow due to a rotating-disk, Philos. Trans. R. Soc. London Ser. A 248, 155–199, 1955.
• Hall, P. An asymptotic investigation of the stationary modes of instability of the boundary layer on a rotating-disk, Proc. Roy. Soc. London Ser. A 406, 93–106, 1986.
• Healey, J. J. On the relation between inviscid and viscous absolute instability of the rotating- disk boundary layer, J. Fluid Mech. 511, 179–199, 2004.
• Huerre, P. and Monkewitz, P. A. Absolute and convective instabilities in free shear layers, J. Fluid Mech. 159, 151–168, 1985.
• Huerre, P. and Monkewitz, P. A. Local and global instabilities in spatially developing flows, Ann. Rev. Fluid Mech. 22, 473–537, 1990.
• Koch, W. Direct resonance in orr-sommerfeld equation, Acta Mech. 58, 11–29, 1986.
• Kupfer, K., Bers, A. and Ram, A. K. The cusp map in the complex-frequency plane for absolute instabilities, Phys. Fluids. 30, 3075–3082, 1987.
• Lingwood, R. J. Absolute instability of the boundary layer on a rotating-disk, J. Fluid Mech. 299, 17–33, 1995.
• Lingwood, R. J. On the application of the briggs’ and steepest-descent method to a boundary- layer flow, Stud. Appl. Math. 98, 213–254, 1997.
• Mackerrel, S. O. A nonlinear asymptotic investigation of the stationary modes of instability of the 3-dimensional boundary layer on a rotating-disk, Proc. Roy. Soc. London Ser. A 413, 497–513, 1987. [26] Malik, M. R. The neutral curve for stationary disturbances in rotating-disk flow, J. Fluid Mech. 164, 275–287, 1986.
• Malik, M. R. and Poll, D. I. A. Effect of curvature on three-dimensional boundary layer stability, AIAA Journal 23, 1362–1369, 1985.
• Malik, M. R., Zang, T. A. and Hussaini, M. Y. A spectral collocation method for the navier- stokes equations, J. Comput. Phys. 61, 64–88, 1984.
• Monkewitz, P. A. The absolute and convective nature of instability in two-dimensional wakes at low reynolds numbers, Phys. Fluids. 31, 999–1006, 1988.
• Pier, B. Finite-amplitude crossflow vortices, secondary instability and transition in the rotating-disk boundary layer, J. Fluid Mech. 487, 315–343, 2003.
• Shanthini, R. Degeneracies of the temporal orr-sommerfeld eigenmodes in plane poiseuille flow, J. Fluid Mech. 201, 13–34, 1989.
• Turkyilmazoglu, M. and Gajjar, J. S. B. On the absolute instability of the attachment-line and swept-hiemenz boundary layers, Theoret. Comput. Fluid Dyn. 13, 57–75, 1999.
• Turkyilmazoglu, M., Gajjar, J. S. B. and Ruban, A. I. The absolute instability of thin wakes in an incompressible/compressible fluid, Theoret. Comput. Fluid Dyn. 13, 91–114, 1999.
• Wilkinson, S. P. and Malik, M. R. Stability experiments in the flow over a rotating-disk, AIAA Journal 23, 588–595, 1985.