- Access by Xinjiang University
Flow induced by a rotating cone: Base flow and convective stability analysis
Phys. Rev. Fluids 4, 084801 – Published 2 August, 2019
DOI: https://doi.org/10.1103/PhysRevFluids.4.084801
Abstract
The steady flow over a rotating cone is investigated theoretically and numerically in order to improve the traditional von Kármán solution by proposing a self-similar correction which is an explicit function of the cone angle. The effect of the correction on the linear stability analysis of the rotating-cone flow is successively investigated through a weakly divergent approach. Both the base flow correction and the results of the stability analysis are validated against dedicated numerical simulations. As for the base flow, the comparison shows a clear improvement obtained by using the proposed correction in comparison with the classical von Kármán solution. As for the stability properties of the flow, the com- parison against the reference simulations shows a good agreement among all the approaches for large azimuthal wave numbers, but a better performance is obtained with the weakly divergent approach for lower wave numbers. The latter approach provides a lower critical Reynolds number than that predicted by parallel theory and, most importantly, changes the interplay between modes I and II with respect to what predicted by the parallel stability calculations. Finally, it is observed that the proposed correction of base flow has a slight effect on the stability analysis of the considered cases, but it may have important effects for low cone angles. Thus, while the classical Kármán solution is appropriate for large cone angles, the proposed correction is recommended for future stability analyses of slender cones.
Physics Subject Headings (PhySH)
Article Text
References (29)
- C. Wu, The three dimensional incompressible laminar boundary layer on a spinning cone, Appl. Sci. Res. 8, 140 (1959).
- F. Kreith, D. Ellis, and J. Giesing, An experimental investigation of the flow engendered by a rotating cone, Appl. Sci. Res. 11, 430 (1963).
- Th. V. Kármán, Über laminare und turbulente Reibung, Z. Angew. Math. Mech. 1, 233 (1921).
- J. C. Y. Koh and J. F. Price, Nonsimilar boundary-layer heat transfer of a rotating cone in forced flow, J. Heat Transfer 89, 139 (1967).
- R. Kobayashi, Linear stability theory of boundary layer along a cone rotating in axial flow, Bull. JSME 24, 934 (1981).
- R. Kobayashi and H. Izumi, Boundary-layer transition on a rotating cone in still fluid, J. Fluid Mech. 127, 353 (1983).
- S. Garrett and N. Peake, The absolute instability of the boundary layer on a rotating cone, Eur. J. Mech. B 26, 344 (2007).
- S. Garrett, Z. Hussain, and S. Stephen, The cross-flow instability of the boundary layer on a rotating cone, J. Fluid Mech. 622, 209 (2009).
- Y. Kohama, Behaviour of spiral vortices on a rotating cone in axial flow, Acta Mech. 51, 105 (1984).
- 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. R. Soc. London 248, 155 (1955).
- P. Hall, An asymptotic investigation of the stationary modes of instability of the boundary layer on a rotating disc, Proc. R. Soc. London A 406, 93 (1986).
- M. R. Malik, The neutral curve for stationary disturbances in rotating-disk flow, J. Fluid Mech. 164, 275 (1986).
- S. Garrett, A. Cooper, J. Harris, M. Özkan, A. Segalini, and P. Thomas, On the stability of von Kármán rotating-disk boundary layers with radial anisotropic surface roughness, Phys. Fluids 28, 014104 (2016).
- C. L. Tien and D. T. Campbell, Heat and mass transfer from rotating cones, J. Fluid Mech. 17, 105 (1963).
- F. Salzberg and S. P. Kezios, Mass transfer from a rotating cone in axisymmetric flow, J. Heat Transfer 87, 469 (1965).
- R. Kappesser, R. Grief, and I. Cornet, Mass transfer to rotating cones, Appl. Sci. Res. 28, 442 (1973).
- M. L. Koosinlin, B. E. Launder, and B. I. Sharma, Prediction of momentum, heat and mass transfer in swirling, turbulent boundary layers, J. Heat Transfer 96, 204 (1974).
- T. Okamoto, M. Yagita, and Y. Kamijima, Experimental investigation on the boundary-layer flow over a rotating cone-cylinder body in a uniform stream, Bull. JSME 19, 930 (1976).
- C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Asymptotic Methods and Perturbation Theory (Springer-Verlag, New York, 2010).
- C.-H. Ling and W. C. Reynolds, Non-parallel flow corrections for the stability of shear flows, J. Fluid Mech. 59, 571 (1973).
- W. S. Saric and A. H. Nayfeh, Nonparallel stability of boundary-layer flows, Phys. Fluids 18, 945 (1975).
- A. Segalini and S. Garrett, On the non-parallel instability of the rotating-sphere boundary layer, J. Fluid Mech. 818, 288 (2017).
- W. G. Cochran, The flow due to a rotating disc, Proc. Cambridge Philos. Soc. 30, 365 (1934).
- http://www.freefem.org.
- F. Viola, E. Pezzica, G. V. V. Iungo, F. Gallaire, and S. Camarri, Flow control of weakly non-parallel flows: Application to trailing vortices, J. Fluid Mech. 822, 342 (2017).
- P. Luchini and A. Bottaro, Adjoint equations in stability analysis, Annu. Rev. Fluid Mech. 46, 493 (2014).
- https://nek5000.mcs.anl.gov.
- K. Kato, T. Kawata, P. H. Alfredsson, and R. J. Lingwood, Investigation of the structures in the unstable rotating-cone boundary layer, Phys. Rev. Fluids 4, 053903 (2019).
- E. Appelquist, S. Imayama, P. H. Alfredsson, P. Schlatter, and R. J. Lingwood, Linear disturbances in the rotating-disk flow: A comparison between results from simulations, experiments and theory, Eur. J. Mech. B 55, 170 (2016).
