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On the computation of transonic leading-edge vortices using the Euler equations

Published online by Cambridge University Press:  21 April 2006

Arthur Rizzi  and
Charles J. Purcell
Arthur Rizzi
Affiliation:
FFA, The Aeronautical Research Institute of Sweden, S-161 11 Bromma, Sweden and Royal Institute of Technology, S-100 44 Stockholm, Sweden
Charles J. Purcell
Affiliation:
ETA Systems, Inc., St Paul, MN 55108, USA
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Abstract

Separation from the leading edge of a delta wing with the subsequent roll-up into a vortex has been simulated in numerical solutions to the Euler equations. Such simulations raise a number of questions that are still outstanding, including the process of inviscid separation from a smooth edge, the role of artificial viscosity in the creation and capturing of vortex sheets, the roll-up mechanism and core features, losses in total pressure, and the stability of the vortical flow structures to three-dimensional disturbances. These matters are discussed in the context of two numerical experiments, both carried out in a sequence of three simulations that starts with a coarse-mesh discretization and ends with a fine mesh of over one million cells. The first experiment is for transonic flow, M∞ = 0.7, α = 10°, around a pure delta wing. The sequence converges to the expected classical steady vortex flow. In the second experiment, transonic flow, M∞ = 0.9, α = 8°, past a twisted cranked-and-cropped delta wing, the sequence does not converge. Instead the crank is observed in the fine-mesh solution to set off an instability in the vortex sheet that causes the vortex to burst into a thin chaotic vortical layer embedded in laminar flow. The mesh sequence suggests that it is the shortest waves resolved that are most unstable, but the energy contained in them comes from the large-scale motion and seems to be small.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 181 , August 1987 , pp. 163 - 195
Copyright
© 1987 Cambridge University Press

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References

Betchov, R. 1965 On the curvature and torsion of an isolated vortex filament. J. Fluid Mech. 22, 471–479.Google Scholar
Brennenstuhl, U. & Hummel, D. 1982 Vortex formation over double-delta-wings. ICAS Paper, 82–6.6.3.
Chorin, A. J. 1982 The evolution of a turbulent vortex. Commun. Math. Phys. 83, 517–535.Google Scholar
Eriksson, L. E. 1982 Generation of boundary-conforming grids around wing-body configurations using transfinite interpolation. AIAA J. 20, 1313–1320.Google Scholar
Fornasier, L. & Rizzi, A. 1985 Comparisons of results from a panel method and an Euler code for cranked delta wing. AIAA Paper 85–4091.
Hall, M. G. 1972 Vortex breakdown. Ann. Rev. Fluid Mech. 4, 195–218.Google Scholar
Hirschel, E. H. & Fornasier, L. 1984 Flowfield and vorticity distribution near wing trailing edges. AIAA paper 84–0421.
Hoeijmakers, H. W. M. 1983 Numerical computation of vortical flows about wings. NLR Rep. MP 83073 U, Nat. Aero. Lab., Amsterdam.Google Scholar
Hoeijmakers, H. W. M. & Rizzi, A. 1984 Vortex-fitted potential solution compared with vortex-captured Euler solution for delta wing with leading edge vortex separation. AIAA Paper 84–2144.
Hoeijmakers, H. W. M. & Vaatstra, W. 1983 A higher-order panel method applied to vortex sheet roll-up. AIAA J. 21, 516–523.Google Scholar
Hoeijmakers, H. W. M., Vaatstra, W. & Verhaagen, N. G. 1983 On the vortex flow over delta and double-delta wings. J. Aircraft 20, 825–832.Google Scholar
Krause, E., Shi, X. G. & Hartwich, P. M. 1983 Computation of leading-edge vortices. AIAA Paper 83–1907.
Leibovich, S. 1978 The structure of vortex breakdown. Ann. Rev. Fluid Mech. 10, 221–246.Google Scholar
Moore, D. W. & Griffith-Jones, R. 1974 The stability of an expanding circular vortex sheet. Mathematika 21, 128–133.Google Scholar
Moore, D. W. 1973 The stability of an evolving two-dimensional vortex sheet. Mathematika 23, 35–44.Google Scholar
Murman, E. M. & Rizzi, A. 1986 Applications of Euler equations to sharp edge delta wings with leading edge vortices. In Proc. AGARD Sym. Appl. Comp. Fluid Dyn. Aero. AGARD-CP-412.
Murman, E., Rizzi, A. & Powell, K. 1985 High resolution solutions of the Euler equations for vortex flows. In Progress and Supercomputing in Computational Fluid Dynamics (ed. S. Abarbanel & E. Murman), pp. 93–113. Boston: Birkhauser.
Powell, K., Murman, E., Perez, E. & Baron, J. 1985 Total pressure loss in vortical solutions of the conical Euler equations. AIAA Paper 85–1701.
Rizzi, A. 1982 Damped Euler-equation method to compute transonic flow around wing-body combinations. AIAA J. 20, 1321–1328.Google Scholar
Rizzi, A. 1985a Euler solutions of transonic vortex flows around the Dillner wing. J. Aircraft 22, 325–328.Google Scholar
Rizzi, A. 1985b Modelling vortex flowfields by supercomputers with super-size memory. Aero J. 89, 149–161.Google Scholar
Rizzi, A. 1987a Multi-cell vortices observed in fine-mesh solutions to the incompressible Euler equations. In Super Computers and Fluid Dynamics (ed. K. Kuwahara). Lecture Notes in Engineering. Springer.
Rizzi, A. 1987b Separation phenomena in 2D and 3D numerical solutions of the Euler equations. In Proc. 7th INRIA Conf. Computing Methods in Appl. Sci. Engng (ed. R. Glowinski et al.). North-Holland.
Rizzi, A. & Bailey, H. E. 1976 Finite volume solution of the Euler equations for steady three-dimensional transonic flow. In Proc. 5th Intl Conf. Num. Methods Fluid Dyn. (ed. A. I. van der Vooren & P. J. Zandbergen). Lecture Notes in Physics, vol. 59. pp. 347–357. Springer.
Rizzi, A. & Eriksson, L. E. 1984 Computation of flow around wings based on the Euler equations. J. Fluid Mech. 148, 45–71.Google Scholar
Rizzi, A. & Eriksson, L. E. 1985 Computation of inviscid incompressible flow with rotation. J. Fluid Mech. 153, 275–312.Google Scholar
Rizzi, A. & Purcell, C. J. 1986a Vortex-stretched flow around a cranked delta wing. J. Aircraft 23, 636–640.Google Scholar
Rizzi, A. & Purcell, C. J. 1986b Disordered vortex flow computed around a cranked delta wing at subsonic speed and high incidence. ICAS Paper No. 86–1.4.1.
Smith, J. H. B. 1984 Theoretical modelling of three-dimensional vortex flows in aerodynamics. Aero J., April 1984, 101–116.
Smith, J. H. B. 1985 Numerical solutions for three-dimensional cases - delta wings. In Test Cases for Inviscid Flow Field Methods. AGARD AR-211.
Snow, J. T. 1978 On inertial instability as related to the multiple-vortex phenomenon. J. Atmos. Sci. 35, 1660–1677.Google Scholar
Thompson, D. H. 1985 A visualization study of the vortex flow around double-delta wings. ARL-AERO-R-165, Melbourne.