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Computation of vortex sheet roll-up in the Trefftz plane

Published online by Cambridge University Press:  21 April 2006

Robert Krasny
Robert Krasny
Affiliation:
Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012, USA Present address: Mathematics Department, The University of Michigan, Ann Arbor, MI 48109, USA.
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Abstract

Two vortex-sheet evolution problems arising in aerodynamics are studied numerically. The approach is based on desingularizing the Cauchy principal value integral which defines the sheet's velocity. Numerical evidence is presented which indicates that the approach converges with respect to refinement in the mesh-size and the smoothing parameter. For elliptic loading, the computed roll-up is in good agreement with Kaden's asymptotic spiral at early times. Some aspects of the solution's instability to short-wavelength perturbations, for a small value of the smoothing parameter, are inferred by comparing calculations performed with different levels of computer round-off error. The tip vortices’ deformation, due to their mutual interaction, is shown in a long-time calculation. Computations for a simulated fuselage-flap configuration show a complicated process of roll-up, deformation and interaction involving the tip vortex and the inboard neighbouring vortices.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 184 , November 1987 , pp. 123 - 155
Copyright
© 1987 Cambridge University Press

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References

Anderson, C. R. 1985 A vortex method for flows with slight density variations. J. Comp. Phys. 61, 417.Google Scholar
Anderson, C. R. 1986 A method of local corrections for computing the velocity field due to a distribution of vortex blobs. J. Comp. Phys. 62, 111.Google Scholar
Baker, G. R. 1979 The ‘Cloud in Cell’' technique applied to the roll up of vortex sheets. J. Comp. Phys. 31, 76.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Birkhoff, G. 1962 Helmholtz and Taylor instability. Proc. Symp. Appl. Math. XIII A.M.S.P. 55.
Carrier, J., Greengard, L. & Rokhlin, V. 1986 A fast adaptive multipole algorithm for particle simulations. Yale Univ. Comp. Sci. Res. Rep. YALEU/DCS/RR-496.
Chorin, A. J. & Bernard, P. S. 1973 Discretization of a vortex sheet, with an example of roll-up. J. Comp. Phys. 13, 423.Google Scholar
DiPerna, R. J. & Majda, A. 1987a Oscillations and concentrations in weak solutions of the incompressible fluid equations. Commun. Math. Phys. 108, 667.Google Scholar
DiPerna, R. J. & Majda, A. 1987b Concentrations in regularizations for 2-D incompressible flow. Commun. Pure Appl. Math. 40, 301.Google Scholar
Donaldson, C. duP., Snedeker, R. S. & Sullivan, R. D. 1974 Calculation of aircraft wake velocity profiles and comparison with experimental measurements. J. Aircraft 11, 547.Google Scholar
Fink, P. T. & Soh, W. K. 1978 A new approach to roll-up calculations of vortex sheets. Proc. R. Soc. Lond. A 362, 195.Google Scholar
Guiraud, J. P. & Zeytounian, R. Kh. 1977 A double-scale investigation of the asymptotic structure of rolled-up vortex sheets. J. Fluid Mech. 79, 93.Google Scholar
Hoeijmakers, H. W. M. 1983 Computational vortex flow aerodynamics. AGARD Conf. Proc. no. 342, paper 18, also NLR MP 83017 U.
Hoeijmakers, H. W. M. 1986 Numerical simulation of vortical flow. In Lecture series Introduction to Vortex Dynamics, Von Karman Inst. Fluid Dyn., Belgium, also NLR MP 86032 U.
Hoeijmakers, H. W. M. & Vaatstra, W. 1983 A higher-order panel method applied to vortex sheet roll-up. AIAA J. 21, 516.Google Scholar
Kaden, H. 1931 Aufwicklung einer unstabilen Unstetgkeitsfläche. Ing. Arch. 2, 140. (English transl., R.A.E. Library Trans. 403.)Google Scholar
Krasny, R. 1986a A study of singularity formation in a vortex sheet by the point vortex approximation. J. Fluid Mech. 167, 65.Google Scholar
Krasny, R. 1986b Desingularization of periodic vortex sheet roll-up. J. Comp. Phys. 65, 292.Google Scholar
Kuwahara, K. & Takami, H. 1973 Numerical studies of two-dimensional vortex motion by a system of point vortices. J. Phys. Soc. Japan 34, 247.Google Scholar
Moore, D. W. 1971 The discrete vortex approximation of a finite vortex sheet. Calif. Inst. Tech. Rep. AFOSR-1804-69.
Moore, D. W. 1974 A numerical study of the roll-up of a finite vortex sheet. J. Fluid Mech. 63, 225.Google Scholar
Moore, D. W. 1975 The rolling up of a semi-infinite vortex sheet. Proc. R. Soc. Lond. A 345, 417.Google Scholar
Moore, D. W. 1976 The stability of an evolving two-dimensional vortex sheet. Mathematika 23, 35.Google Scholar
Moore, D. W. & Saffman, P. G. 1973 Axial flow in laminar trailing vortices. Proc. R. Soc. Lond. A 333, 491.Google Scholar
Murman, E. M. & Stremel, P. M. 1982 A vortex wake capturing method for potential flow calculations. AIAA paper 82-0947.
Overman II, E. A. & Zabusky, N. I. 1982 Evolution and merger of isolated vortex structures. Phys. Fluids 25, 1297.Google Scholar
Pullin, D. I. 1978 The large-scale structure of unsteady self-similar rolled-up vortex sheets. J. Fluid Mech. 88, 401.Google Scholar
Pullin, D. I. & Phillips, W. R. C. 1981 On a generalization of Kaden's problem. J. Fluid Mech. 104, 45.Google Scholar
Rosenhead, L. 1931 The formation of vortices from a surface of discontinuity. Proc. R. Soc. Lond. A 134, 170.Google Scholar
Rott, N. 1956 Diffraction of a weak shock with vortex generation. J. Fluid Mech. 1, 111.Google Scholar
Saffman, P. G. & Baker, G. R. 1979 Vortex interactions. Ann. Rev. Fluid Mech. 11, 95.Google Scholar
Smith, J. H. B. 1986 Vortex flows in aerodynamics. Ann. Rev. Fluid Mech. 18, 221.Google Scholar
Stein, E. M. & Weiss, G. 1971 Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press.
Takami, H. 1964 A numerical experiment with discrete vortex approximation with reference to the rolling up of a vortex sheet. Dept. Aero & Astron., Stanford Univ. SUDAER 202.
Tryggvason, G. 1987 Simulation of vortex sheet roll-up by vortex methods. J. Comp. Phys. (to appear).Google Scholar
Weston, R. P. & Liu, C. H. 1982 Approximate boundary condition procedure for the two-dimensional numerical solution of vortex wakes. AIAA paper 82-0951.
Westwater, F. L. 1935 Rolling up of the surface of discontinuity behind an aerofoil of finite span. Aero. Res. Counc. R & M. 1692.