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Multigrid multiblock hovering rotor solutions

Published online by Cambridge University Press:  03 February 2016

C. B. Allen
C. B. Allen*
Affiliation:
Department of Aerospace Engineering, University of Bristol, Bristol, UK
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Abstract

The effect of multigrid acceleration implemented within an upwind-biased Euler method for hovering rotor flows is presented. Previous work has considered multigrid convergence for structured single block rotor solutions. However, for forward flight simulation a multiblock approach is essential and, hence, the flow-solver has been extended to include multigrid acceleration within a multiblock solver. The requirement to capture the vortical wake development over several turns means a long numerical integration time is required for hovering rotors, and the solution (wake) away from the blade is significant. Hence, the solution evolution and convergence is different to a fixed wing case where convergence depends primarily on propagating errors away from the surface as quickly as possible, and multigrid acceleration is shown here to be less effective for hovering rotor flows. Previous single block simulations demonstrated that a simple multigrid V-cycle was the most effective, smoothing in the decreasing mesh density direction only, with a relaxed trilinear prolongation operator. This is also shown to be the case for multiblock simulations. Results are presented for multigrid computations with 2, 3, and 4, mesh levels, and a CPU reduction of approximately 80% is demonstrated for 4 mesh levels.

Type
Research Article
Information
The Aeronautical Journal , Volume 108 , Issue 1083 , May 2004 , pp. 255 - 261
Copyright
Copyright © Royal Aeronautical Society 2004 

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References

1. Fedorenko, R.P. The speed of convergence of one iterative process, Zh Vych Mat, 3, (4), 1964, pp 559–564 (USSR Comp Math and Math Phys, 1964, (4), pp 227–235).Google Scholar
2. Bakhvalov, N.S. On the convergence of a relaxation method with natural constraints of the elliptic operator, Zh Vych Mat, 1966, 5, (6), pp 861–885. (USSR Comp Math and Math Phys, 1966, (6), pp 101–135).Google Scholar
3. Brandt, A. Multi-level adaptive solutions to boundary-value problems, Math Comp, 1977, 31, (138), pp 333–390.Google Scholar
4. Brandt, A. Guide to Multigrid Development, Multigrid Methods (Hackbusch, and Trottenberg, , Eds), Springer-Verlag, Lecture Notes in Mathematics, 960, 1982.Google Scholar
5. Jameson, A. Transonic flow calculations, Princeton University, Report MAE 1751, 1984.Google Scholar
6. Jameson, A. Time-dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings, AIAA Paper 91-1596, 1991.Google Scholar
7. Ni, R.H. A multiple-grid scheme for solving the Euler equations, AIAA J, 1982, 20, (11), pp 1565–1571.Google Scholar
8. Arnone, A., Liou, M-S. and Povinelli, L.A. Multigrid time-accurate integration of Navier-Stokes equations, AIAA Paper 93-3361, 1993.Google Scholar
9. Kroll, N. Computation of the flo fields of propellers and hovering rotors using Euler equations, Paper 28, 12th European Rotorcraft Forum, Garmisch-Partenkirchen, Germany, September 1986.Google Scholar
10. Boniface, J.C. and Sides, J. Numerical simulation of steady and unsteady Euler flows around multibladed helicopter rotors, Paper C10, 19th European Rotorcraft Forum, Cernobbio (Como), Italy, September 1993.Google Scholar
11. Pahlke, K. and Raddatz, J. 3D Euler methods for multibladed rotors in hover and forward flight, Paper 20, 19th European Rotorcraft Forum, Cernobbio (Como), Italy, September 1993.Google Scholar
12. Raddatz, J. and Pahlke, K. 3D Euler calculations of multibladed rotors in hover: Investigations of the wake capturing properties, 75th AGARD Fluid Dynamics Panel Meeting and Symposium on Aerodynamics and Aeroacoustics of Rotorcraft, Berlin, Germany, October 1994.Google Scholar
13. Raddatz, J. and Rouzard, O. Calculations of multibladed rotors in hover using 3D Euler methods of DLR and Onera, Paper 11, 21st European Rotorcraft Forum, St Petersburg, Russia, August 1995.Google Scholar
14. Boniface, J.C. and Sides, J. Extension and improvement of existing Euler code of ONERA for multibladed rotors in hover, HELISHAPE Technical Report, 1995.Google Scholar
15. Sankar, L.N., Wake, B.E. and Lekoudis, S.G. Solution of the unsteady Euler equations for fixed and rotating wing configurations, AIAA J Aircr, April 1986, 23, (4), pp 283–289.Google Scholar
16. Sankar, L.N., Wake, B.E. and Lekoudis, S.G., Computation of rotor blade flows using the Euler equations, AIAA J of Aircr, July 1986, 23, (7), pp 582–588.Google Scholar
17. Strawn, R.C. and Barth, T.J. A finite-volume Euler solver for computing rotary-wing aerodynamics on unstructured meshes, In Proceedings of the 48th Annual American Helicopter Society Forum, Washington, DC, June 1992.Google Scholar
18. Allen, C.B. The effect of grid topology and density on inviscid hovering rotor solutions, I Mech E J of Aero Eng, Part G7, 1999.Google Scholar
19. Srinivasan, G.R., Baeder, J.D., Obayashi, S. and McCroskey, W.J. Flowfield of a lifting rotor in hover. A Navier-Stokes simulation, AIAA J Aircr, October 1992, 30, (10), pp 2371–2377.Google Scholar
20. Wake, B.E. and Egolf, T.A. Implementation of a rotary-wing Navier-Stokes solver on a massively parallel computer, AIAA J Aircr, January 1991, 29, (1), pp 58–67.Google Scholar
21. Zhong, B. and Qin, N. Non-inertial multiblock Navier-Stokes calculation for hovering rotor flowfields using high order upwind scheme, Proceedings Royal Aeronautical Society Aerodynamics Conference, London, April 2000.Google Scholar
22. Van-Leer, B. Flux-vector splitting for the Euler equations, Lecture Notes in Physics, 1982, 170, pp 507–512.Google Scholar
23. Allen, C.B. Multigrid acceleration of an upwind Euler code for hovering rotor flows, Aeronaut J, September 2001, 105, (1051), pp 517–524.Google Scholar
24. Allen, C.B. Multigrid convergence of inviscid fixed-and rotary-wing flows, Int J for Numerical Methods in Fluids, 2002, 39, (2), pp 121–140.Google Scholar
25. Allen, C.B. Time-stepping and grid effects on convergence of inviscid rotor flows, AIAA paper 2002-2817, proceedings 20th Applied Aerodynamics Conference, St Louis, June 2002.Google Scholar
26. Allen, C.B. Convergence of steady and unsteady inviscid hovering rotor solutions, Int J for Numerical Methods in Fluids, 2003, 41, (9), pp 931–949.Google Scholar
27. Parpia, I. H. Van-Leer flux-vector splitting in moving coordinates, AIAA J, January 1988, 26, pp 113–115.Google Scholar
28. Anderson, W.K., Thomas, J.L. and Van-Leer, B., Comparison of finite volume flux vector splittings for the Euler equation, AIAA J, September 1986, 24, pp 1453–1460.Google Scholar
29. Jameson, A., Schmidt, W. and Turkel, E. Numerical solution of the Euler equations by finite-volume methods using Runge-Kutta time-stepping schemes’, AIAA Paper 81-1259, 1981.Google Scholar
30. Turkel, E. and Van-Leer, B. Runge-Kutta methods for partial differential equations, ICASE Report, 1983.Google Scholar
31. Allen, C.B. ‘CHIMERA volume grid generation within the EROS code, I Mech E J Aerosp Eng, Part G, 2000.Google Scholar
32. Gordon, W.J. and Hall, C.A. Construction of curvilinear coordinate systems and applications of mesh generation, Int J Numerical Methods in Eng, 7, 1973, pp 461–477.Google Scholar
33. Thompson, J.F. A general three dimensional elliptic grid generation system on a composite block-structure, Computer Methods in Appl Mech and Eng, 64, 1987, pp 377–411.Google Scholar
34. Caradonna, F.X. and Tung, C. Experimental and analytical studies of a model helicopter rotor in hover, NASA TM-81232, September 1981.Google Scholar
35. Grasso, F. and Marini, M. Multigrid techniques for hypersonic viscous flows, AIAA J, 1993, 31, pp 1729–1731.Google Scholar
36. Grasso, F. and Marini, M. Solutions of hypersonic viscous flows with total variation diminishing Multigrid techniques, Computers and Fluids, 1995, 24, pp 571–592.Google Scholar
37. Dick, E. Multigrid formulation of polynomial flux-difference splitting for steady Euler equations, JCP, 1990, 91, pp 161–173.Google Scholar
38. Dick, E. Multigrid methods for steady Euler and Navier-Stokes equations based on polynomial flux-difference splitting, In Multigrid Methods II, Birkhauser Verlag, 1–20, 1991.Google Scholar
39. Satofuka, N. and Morinishi, K. Method of lines approach to CFD using rational Runge-Kutta time-stepping scheme, In Frontiers of Computational Fluid Dynamics, Caughey, and Hafez, (Eds), 1995, pp 189–213.Google Scholar
40. Mavriplis, D.J. Three-dimensional Multigrid for the Euler equations, AIAA J, 1992, 30, pp 1753–1761.Google Scholar