Abstract
A two dimensional, unsteady, linearized Euler solver has been developed, and applied to both flutter and forced response problems. Solutions are obtained at a single frequency, with the time derivatives ∂Q/∂t replaced by −iωQ. The unsteady solver is derived from an existing steady flow Euler solver that uses adaptive triangular grids. The solver is not restricted to blade row geometries. The solution technique is the false time marching of Ni, used in conjunction with a Runge-Kutta scheme. As with most steady flow Euler solvers, shocks are captured, rather than fitted. The inlet and outlet boundary conditions are the two dimensional single frequency non-reflective boundary conditions due to Giles. For flutter problems, the grid moves with the blade: grid points on the blade surface remain fixed to the blade, and the motion of interior grid points is computed using a Laplacian smoother. Results are shown for three flutter calculations and one forced response calculation.
Potential methods are clearly more efficient than linearized Euler methods, but are restricted to shock free flows, or to flows with weak shocks and shock geometries simple enough to permit shock fitting. This and other investigations have shown that Euler methods can produce results comparable to potential methods. For transonic flows, the unsteady force applied to a blade by an oscillating shock — the “shock foot force” — can depend significantly on the way the shock is modeled. Improvements in the way unsteady Euler solvers treat shocks are required, possibly even extending to making the grid oscillate with the shock. Once the issue of the correct treatment of shocks can be resolved, Euler methods will be able to produce useful solutions in regimes inaccessible to potential methods.
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References
J. T. Batina. Implicit flux-split Euler schemes for unsteady aerodynamic analysis involving unstructured dynamic meshes. AIAA Paper 90–0936, 1990.
A. Bolcs and T. H. Fransson. Aeroelasticity in turbomachines: Comparison of theoretical and experimental cascade results. Communication du Laboratoire de Thermique Appliquée et de Turbomachines Nr. 13, L’École Polytechnique Fédérale de Lausanne, 1986.
G. Dahlquist and A. Bjorck. Numerical Methods Prentice-Hall, New Jersey, 1974.
M. B. Giles. UNSFLO: A numerical method for unsteady in-viscid flow in turbomachinery. Computational Fluid Dynamics Laboratory Report CFDL-TR-86–6, MIT, 1986.
M. B. Giles. Non-reflecting boundary conditions for the Euler equations. Computational Fluid Dynamics Laboratory Report CFDL-TR-88–1, MIT, 1988.
M. B. Giles. Numerical methods for unsteady turbomachinery flow. In Numerical Methods for Flows in Turbomachinery,VKI Lecture Series 1989–06, 1989.
K. C. Hall. A Linearized Euler Analysis of Unsteady Flows in Turbomachinery PhD thesis, M.I.T., 1987.
K. C. Hall. Calculation of unsteady flows in turbomachines using the linearized Euler equations. AIAA Journal,27(6):777–787, 1989.
D. G. Holmes and S. D. Connell. Solution of the 2D NavierStokes equations on unstructured, adaptive grids. AIAA 9th Computational Fluid Dynamics Conference, AIAA Paper 891932-CP, 1989.
L. He. An Euler solver for unsteady flows around oscillating blades. ASME Paper 89-GT-279, 1989.
D. G. Holmes, S. H Lamson, and S. D. Connell. Quasi-3D solutions for transonic, inviscid flows by adaptive triangulation. ASME Paper 88-GT-83, 1988.
A. Jameson, T. J. Baker, and N. P. Weatherill. Calculation of inviscid transonic flow over a complete aircraft. AIAA paper 86–0103, 1986.
A. Jameson, W. Schmidt, and E. Turkel. Numerical solutions of the Euler equations by finite-volume methods using Runge-Kutta time-stepping scheme. AIAA paper 81–1259, 1981.
L. M. Landau. Fluid Mechanics Pergamon Press, London, 1959.
D. R. Lindquist and M. B. Giles. On the validity of linearized unsteady Euler equations with shock capturing. AIAA 10th Computational Fluid Dynamics Conference, AIAA Paper 91–1598-CP, 1991.
D. J. Mavripilis, A. Jameson, and L. Martinelli. Multigrid solution of the Navier-Stokes equations on triangular meshes. AIAA Paper AIAA-89–0120, 1989.
R.-H. Ni. Non-Stationary Aerodynamics of Flat Plate Cascades in Compressible Flow PhD thesis, Stevens Institute of Technology, 1974.
IL-H. Ni and J. Sisto. Non-stationary aerodynamics of flat plate cascades in compressible flow. ASME Paper 75-GT-5, 1975.
L. D. G. Siden. Numerical Simulation of Viscous Compressible Flows Applied to Turbomachinery Blade Flutter PhD thesis, Chalmers University of Technology, Goteborg, Sweden, 1991.
J. M. Verdon. Further developments in the aerodynamic analysis of unsteady supersonic cascades, parts 1 & 2. ASME Paper 77GT-44 & 45, 1977.
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© 1993 Springer-Verlag New York, Inc.
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Holmes, D.G., Chuang, H.A. (1993). 2D Linearized Harmonic Euler Flow Analysis for Flutter and Forced Response. In: Atassi, H.M. (eds) Unsteady Aerodynamics, Aeroacoustics, and Aeroelasticity of Turbomachines and Propellers. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9341-2_11
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DOI: https://doi.org/10.1007/978-1-4613-9341-2_11
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