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On a class of unsteady three-dimensional Navier—Stokes solutions relevant to rotating disc flows: threshold amplitudes and finite-time singularities

Published online by Cambridge University Press:  26 April 2006

Philip Hall ,
P. Balakumar  and
D. Papageorgiu
Philip Hall
Affiliation:
Mathematics Department, University of Manchester, Oxford Road, M139TL, UK
P. Balakumar
Affiliation:
High Technology Corporation, Hampton, VA, USA
D. Papageorgiu
Affiliation:
New Jersey Institute of Technology, Newark. NJ. USA
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Abstract

A class of ‘exact’ steady and unsteady solutions of the Navier—Stokes equations in cylindrical polar coordinates is given. The flows correspond to the motion induced by an infinite disc rotating in the (x, y)-plane with constant angular velocity about the z-axis in a fluid occupying a semi-infinite region which, at large distances from the disc, has velocity field proportional to (x, — y,O) with respect to a Cartesian coordinate system. It is shown that when the rate of rotation is large Kármán's exact solution for a disc rotating in an otherwise motionless fluid is recovered. In the limit of zero rotation rate a particular form of Howarth's exact solution for three-dimensional stagnation-point flow is obtained. The unsteady form of the partial differential system describing this class of flow may be generalized to time-periodic flows. In addition the unsteady equations are shown to describe a strongly nonlinear instability of Kármán's rotating disc flow. It is shown that sufficiently large perturbations lead to a finite-time breakdown of that flow whilst smaller disturbances decay to zero. If the stagnation point flow at infinity is sufficiently strong the steady basic states become linearly unstable. In fact there is then a continuous spectrum of unstable eigenvalues of the stability equations but, if the initial-value problem is considered, it is found that, at large values of time, the continuous spectrum leads to a velocity field growing exponentially in time with an amplitude decaying algebraically in time.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 238 , May 1992 , pp. 297 - 323
Copyright
© 1992 Cambridge University Press

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References

Balakumar, P., Hall, P. & Malik, M. 1991 The nonparallel receptivity for rotating disc flows. Theor. Comp. Fluid Dyn. (to appear).Google Scholar
Banks, W. H. H. & Zaturska, A. B. 1979 The collision of unsteady laminar boundary layers. J. Engng Maths 13, 193.Google Scholar
Banks, W. H. H. & Zaturska, A. B. 1989 Eigenvalues at a three-dimensional stagnation point. Acta Mechanica 78, 39.Google Scholar
Bassom, A. P. & Gajjar, J. S. B. 1988 Non-stationary cross-flow vortices in three-dimensional boundary layer flows. Proc. R. Soc. Lond.. A 417, 179.Google Scholar
Bodonyi, R. J. & Ng, B. 1984 On the stability of the similarity solution for swirling flow above an infinite rotating disc. J. Fluid Mech. 144, 311.Google Scholar
Bodonyi, R. J. & Stewartson, K. 1977 The unsteady laminar boundary layer on a disc in counter-rotating fluids. J. Fluid Mech. 79, 669.Google Scholar
Davey, A. 1961 Boundary-layer flow at a saddle point of attachment. J. Fluid Mech. 4, 593.Google Scholar
Federvov, B. I., Plavnik, G. Z., Prokholov, I. V. & Zhukhovitskii, L. G. 1976 Transitional flow conditions on a rotating disc. J. Engng Phys. 31, 1448.Google Scholar
Gregory, N., Stuart, J. T. & Walker, W. S. 1955 On the stability of three-dimensional boundary layers with applications to the flow due to a rotating disc. Phil. Trans. R. Soc. Lond. A 248, 155.Google Scholar
Hall, P. 1986 An asymptotic investigation of the stationary modes of instability of the boundary layer on a rotating disc. Proc. R. Soc. Lond. A 406, 93.Google Scholar
Hall, P. & Smith, F. T. 1991 On strongly nonlinear vortex/wave interactions in boundary layer transition. J. Fluid Mech. 227, 641–666.Google Scholar
Howarth, L. 1951 The boundary layer in three-dimensional flow. Part II. The stagnation point. Phil. Mag. 7 (62), 1433.Google Scholar
Kármán, T. 1921 Über laminare und turbulente reibung. Z. angew. Math. Mech. 1, 233.Google Scholar
MacKerrell, S. O. 1987 A nonlinear asymptotic investigation of the three-dimensional boundary layer on a rotating disc. Proc. R. Soc. Lond. A 413, 497.Google Scholar
Malik, M. 1986 The neutral curve for stationary disturbances in rotating disc flow. J. Fluid Mech. 164, 275.Google Scholar
Schofield, D. & Davey, A. 1967 Dual solutions of the boundary layer of attachment. J. Fluid Mech. 30, 809.Google Scholar
Stuart, J. T. 1988 Nonlinear Euler partial differential equations: singularities in their solution. In Symposium to Honour C. C. Lin. World Science.