Nonlinear finite-element analysis of a two-dimensional panel oscillating in supersonic flow

Abstract

In this study, the total Lagrangian formulation (TLF) is implemented in a finite-element structural solver to analyze the panel flutter problem, which involves large structural deformations. To accurately compute the aerodynamic loads acting on the panel while considering the panel’s deformations, the structural solver is coupled with a finite-element computational fluid dynamics (CFD) solver, which adopts the arbitrary Lagrangian–Eulerian (ALE) formulation to solve the unsteady Euler equations. The coupled solver is then applied to analyze the flutter behavior of a panel in the supersonic flow regime. Finally, the results are compared with those reported in the literature, which rely on the von Karman nonlinear theory based on the small strain assumption, as well as other studies using the TLF within a finite-volume framework. This study demonstrates that the panel undergoes limit-cycle oscillations (LCOs), with generally good agreement between the findings of this study and the existing literature in predicting the amplitude and frequency of the LCOs. However, discrepancies in amplitude are observed between the current results and those obtained using von Kármán theory, with increasing divergence as the amplitudes approach the panel thickness. In addition, for the case of a free-stream Mach number of 1.8, results obtained using the TLF method within a finite-volume framework indicate that the panel oscillates in higher modes (seventh mode). In contrast, both this study, employing TLF within a finite-element framework, and those employing von Kármán theory predict oscillation as a combination of the first and second modes. These findings highlight the need for further investigation, particularly with experimental results, which could provide valuable insights into the actual behavior of the panel under such conditions.

Keywords

Panel flutter total Lagrangian formulation (TLF)finite-element method (FEM)limit-cycle oscillations (LCOs)aeroelasticity

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References

Aeroelastic Flutter–Flow Physics and Computation Lab, https://engineering.jhu.edu/fsag/research/aeroelastic-flutter/\#:œ:text=Aeroelastic\%20Flutter-,Aeroelastic\%20Flutter,wide\%20range\%20of\%20engineering\%20fields.

Dowell

. A modern course in aeroelasticity. Cham: Springer, 2015.

Dowell

Voss

. Theoretical and experimental panel flutter studies in the mach number range 1.0 to 5.0. AIAA J 1965; 3(12): 2292–2304.

von Kármán

. Strength problems in engineering. In: Klein

Meyer

(eds) Encyclopaedia of mathematical sciences including their applications. Vol. IV: Mechanics. Leipzig/Berlin: B.G. Teubner Verlag, pp. 311–385.

Ciarlet

. Plates and junctions in elastic multi-structures: an asymptotic analysis. Cham: Springer, 1991.

Bathe

Ramm

Wilson

. Finite element formulations for large deformation dynamic analysis. Int J Numer Meth Eng 1975; 9(2): 353–386.

Lighthill

. Oscillating airfoils at high mach number. J Aeronaut Sci 1953; 20(6): 402–406.

Dowell

. Nonlinear oscillations of a fluttering plate. AIAA J 1966; 4(7): 1267–1275.

Mei

. A finite-element approach for nonlinear panel flutter. AIAA J 1977; 15(8): 1107–1110.

10.

Dowell

Bliss

. New look at unsteady supersonic potential flow aerodynamics and piston theory. AIAA J 2013; 51(9): 2278–2281.

11.

Ganji

Dowell

. Panel flutter prediction in two dimensional flow with enhanced piston theory. J Fluid Struct 2016; 63: 97–102.

12.

Dowell

. Nonlinear oscillations of a fluttering plate. II. AIAA J 1967; 5(10): 1856–1862.

13.

Davis

Bendiksen

. Transonic panel flutter. In: 34th structures, structural dynamics and materials conference, La Jolla, CA, 19 April–22 April 1993. https://arc.aiaa.org/doi/pdf/10.2514/6.1993-1476

14.

Hejranfar

Azampour

. Simulation of 2d fluid–structure interaction in inviscid compressible flows using a cell-vertex central difference finite volume method. J Fluid Struct 2016; 67: 190–218.

15.

Gordnier

Visbal

. Development of a three-dimensional viscous aeroelastic solver for nonlinear panel flutter. J Fluid Struct 2002; 16(4): 497–527.

16.

Gordnier

Fithen

. Coupling of a nonlinear finite element structural method with a Navier–Stokes solver. Comput Struct 2003; 81(2): 75–89.

17.

Alder

. Development and validation of a fluid–structure solver for transonic panel flutter. AIAA J 2015; 53(12): 3509–3521.

18.

Cinquegrana

Vitagliano

. Validation of a new fluid—structure interaction framework for non-linear instabilities of 3d aerodynamic configurations. J Fluid Struct 2021; 103: 103264.

19.

Cinquegrana

Vitagliano

. Non-linear panel instabilities at high-subsonic and low supersonic speeds solved with strongly coupled CIRA FSI framework. Int J Non Linear Mech 2021; 129: 103643.

20.

Bathe

. Finite element procedures. New York: Prentice Hall, 2006.

21.

Reddy

. An introduction to nonlinear finite element analysis: with applications to heat transfer, fluid mechanics, and solid mechanics. Oxford: Oxford University Press, 2014.

22.

Newmark

. A method of computation for structural dynamics. J Eng Mech Div 1959; 85(3): 67–94.

23.

Reddy

. An introduction to finite element method. New York: McGraw-Hill Education, 2006.

24.

Belytschko

Liu

Moran

, et al. Nonlinear finite elements for continua and structures. Hoboken, NJ: Wiley, 2013.

25.

Kuhl

Hulshoff

de Borst

. An arbitrary Lagrangian Eulerian finite-element approach for fluid–structure interaction phenomena. Int J Numer Meth Eng 2003; 57(1): 117–142.

26.

Akargün

. Least-squares finite element solution of Euler equations with adaptive mesh refinement. Master’s Thesis, Middle East Technical University, Ankara, Turkey, 2012.

27.

Hughes

Franca

Hulbert

. A new finite element formulation for computational fluid dynamics: Viii. the galerkin/least-squares method for advective-diffusive equations. Comput Method Appl M 1989; 73(2): 173–189.

28.

Ghobrial

. Large deformation analysis of fluid-structure interaction problems. Master’s Thesis, Cairo University, Giza, Egypt, 2021.

29.

Donea

Huerta

Ponthot

, et al. Arbitrary Lagrangian–Eulerian methods. In: Stein

de Borst

Hughes

TJR

(eds) Encyclopedia of computational mechanics. Hoboken, NJ: John Wiley, 2004, pp. 413–437.

30.

Jeong

Seong

. Comparison of effects on technical variances of computational fluid dynamics (CFD) software based on finite element and finite volume methods. Int J Mech Sci 2014; 78: 19–26.