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Penetration of a blade into a vortex core: vorticity response and unsteady blade forces

Published online by Cambridge University Press:  26 April 2006

J. S. Marshall  and
J. R. Grant
J. S. Marshall
Affiliation:
Department of Mechanical Engineering and Iowa Institute of Hydraulic Research, The University of Iowa, Iowa City, IA 52242, USA
J. R. Grant
Affiliation:
Naval Undersea Warfare Center, Building 1302, Newport, RI 02841, USA
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Abstract

Numerical calculations are performed for the problem of penetration into a vortex core of a blade travelling normal to the vortex axis, where the plane formed by the blade span and the direction of blade motion coincides with the normal plane of the vortex axis at the point of penetration. The calculations are based on a computational method, applicable for unsteady three-dimensional flow past immersed bodies, in which a collocation solution of the vorticity transport equation is obtained on a set of Lagrangian control points. Differences between this method and other Lagrangian vorticity-based methods in the literature are discussed. Lagrangian methods of this type are particularly attractive for problems of unsteady vortex-body interaction, since control points need only be placed on the surface of the body and in regions of the flow with non-negligible vorticity magnitude. The computations for normal blade-vortex interaction (BVI) are performed for an inviscid fluid and focus on the relationship between the vortex core deformation due to penetration of the blade into the vortex ambient position and the resulting unsteady pressure field and unsteady force acting on the blade. Computations for cases with different vortex circulations are performed, and the accuracy of an approximate formulation using rapid distortion theory is assessed by comparison with the full computational results for unsteady blade force. The force generated from blade penetration into the vortex ambient position is found to be of a comparable magnitude to various other types of unsteady BVI forces, such as that due to cutting of the vortex axial flow.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 306 , 10 January 1996 , pp. 83 - 109
Copyright
© 1996 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.
Ahmadi, A. R. 1986 An experimental investigation of blade-vortex interaction at normal incidence. J. Aircraft 23, 47–55.Google Scholar
Amiet, R. K. 1986 Airfoil gust response and the sound produced by airfoil-vortex interaction. J. Sound Vib. 107, 487–506.Google Scholar
Amiet, R. K., Simonich, J. C. & Schlinker, R. H. 1990 Rotor noise due to atmospheric turbulence ingestion. Part II. Aeroacoustic results. J. Aircraft 27, 15–22.Google Scholar
Anderson, C. & Greengard, C. 1985 On vortex methods. SIAM J. Numer. Anal. 22, 413–440.Google Scholar
Beale, J. T. 1986 On the accuracy of vortex methods. In Proc. Workshop on Comput. Fluid Dyn. & Reacting Gas Flows, IMA, University of Minnesota.
Beale, J. T. & Majda, A. 1982 Vortex methods. Part I. Convergence in three dimensions. Math. Comput. 39, 1–27.Google Scholar
Beck, J. V., Blackwell, B. & St. Clair, C. R. 1985 Inverse Heat Conduction. John Wiley & Sons.
Butler, S. F. J. 1953 A note on Stoke's stream function for motion with a spherical boundary. Proc. Comb. Phil. Soc. 49, 169–174.Google Scholar
Cary, C. M. 1987 An experimental investigation of the chopping of helicopter main rotor tip vortices by the tail rotor. Part II. High speed photographic study. NASA CR-177457.
Choquin, J. P. & Lucquin-Desreux, B. 1988 Accuracy of a deterministic particle method for Navier-Stokes equations. Intl J. Numer. Meth. Fluids 8, 1439–1458.Google Scholar
Gradshteyn, I. S. & Ryzhik, I. M. 1980 Tables of Integral, Series and Products Academic.
Greengard, L. & Rokhlin, V. 1987 A fast algorithm for particle simulations. J. Comput. Phys. 73, 325–348.Google Scholar
Helmholtz, H. 1858 Uber Integrale der hydrodynamischen Gleichungen welche den Wirbelbewegungen entsprechen. Crelles J. 55, 25.Google Scholar
Hess, J. & Smith, A. 1967 Calculation of potential flows about arbitrary bodies. Prog. Aeronaut. Sci. 8, 1–138.Google Scholar
Howe, M. S. 1988 Contributions to the theory of sound production by vortex-airfoil interaction, with application to vortices with finite axial velocity defect. Proc. R. Soc. Lond. A 420, 157–182.Google Scholar
Howe, M. S. 1989 On unsteady surface forces, and sound produced by the normal chopping of a rectilinear vortex. J. Fluid Mech. 206, 131–153.Google Scholar
Johnston, R. T. & Sullivan, J. P. 1992 Unsteady wing surface pressures in the wake of a propellor. AIAA Paper 92-0277.
Knio, O. M. & Ghoniem, A. F. 1990 Numerical study of a three-dimensional vortex method. J. Comput. Phys. 86, 75–106.Google Scholar
Krishnamoorthy, S. 1993 An experimental study of vortex response during cutting by a blade or cylinder. 1 thesis, Florida Atlantic University, Boca Raton, Florida.
Krishnamoorthy, S. & Marshall, J. S. 1994 An experimental investigation of ‘vortex shocks’. Phys. Fluids 6, 3737–3741.Google Scholar
Leonard, A. 1985 Computing three-dimensional incompressible flows with vortex elements. Ann. Rev. Fluid Mech. 17, 523–559.Google Scholar
Leverton, J. W., Pollard, J. S. & Wills, C. R. 1977 Main rotor wake/tail rotor interaction. Vertica 1, 213–221.Google Scholar
Lundgren, T. S. & Ashurst, W. T. 1989 Area-varying waves on curved vortex tubes with application to vortex breakdown. J. Fluid Mech. 200, 283–307.Google Scholar
Marshall, J. S. 1991 A general theory of curved vortices with circular cross-section and variable core radius. J. Fluid Mech. 229, 311–338.Google Scholar
Marshall, J. S. 1994 Vortex cutting by a blade. Part I. General theory and a simple solution. AIAA J. 32, 1145–1150.Google Scholar
Marshall, J. S. & Grant, J. R. 1994 Evolution and breakup of vortex rings in straining and shearing flows. J. Fluid Mech. 273, 285–312.Google Scholar
Marshall, J. S. & Yalamanchili, R. 1994 Vortex cutting by a blade. Part II. Computations of the vortex response. AIAA J. 32, 1428–1436.Google Scholar
Novikov, E. A. 1983 Generalized dynamics of three-dimensional vortical singularities (vortons). Sov. Phys. JETP 57, 566–569.Google Scholar
Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T. 1992 Numerical Recipes, 2nd edn. Cambridge University Press.
Saffman, P. G. 1970 The velocity of viscous vortex rings. Stud. Appl. Maths 49, 371–380.Google Scholar
Schlinker, R. H. & Amiet, R. K. 1983 Rotor-vortex interaction noise. AIAA Paper 83-0720.
Simonich, J. C., Amiet, R. K. & Schlinker, R. H. 1990 Rotor noise due to atmospheric turbulence ingestion. Part I: Fluid mechanics. J. Aircraft 27, 7–14.Google Scholar
Uhlman, J. S. 1992 An integral equation formation of the equations of motion of an incompressible fluid. Naval Undersea Warfare Center, Division Newport, Tech. Rep. 10086.Google Scholar
Uhlman, J. S. & Grant, J. R. 1993 A new method for implementation of boundary conditions in the discrete vortex element method. Proc. ASME Fluids Engng Conf., June, Washington, DC.
Weigand, A. 1993 The response of a vortex ring to a transient, spatial cut. PhD Dissertation, University of California, San Diego.
Winckelmans, G. S. & Leonard, A. 1993 Contributions to vortex particle methods for the computation of three-dimensional incompressible unsteady flows. J. Comput. Phys. 109, 247–273.Google Scholar