Aeroelastic stability of plate interacting with a flowing fluid

S. A. Bochkarev ; S. V. Lekomtsev

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 20 (2016) no. 3, pp. 552-566.

Résumé

The paper presents the results of a numerical study of the dynamic behavior of the deformable plate interacting both with the external supersonic gas flow and the internal fluid flow. The constitutive relations describing the behavior of ideal compressible fluid in the case of small perturbations are written in terms of the perturbation velocity potential and transformed using the Bubnov–Galerkin method. The aero- and dynamic pressures are calculated based on the quasi-static aerodynamic theory. The strains in the plate evaluated following the Timoshenko hypotheses. A mathematical formulation of the dynamic problem of elastic structure is developed using the variational principle of virtual displacements, which takes into account the work done by the inertia forces, aerodynamic and hydrodynamic pressures. Calculation of complex eigenvalues of the coupled system of two equations is performed using an algorithm based on implicitly restarted Arnoldi method. The stability criterion is based on an analysis of the complex eigenvalues of system of two equations obtained for increasing flow or gas velocity. The reliability of the obtained numerical solution has been estimated by comparing it with the available theoretical data. A few numerical examples were considered to demonstrate the existence of different types of instability depending on the velocities of fluid or gas flow, combinations of kinematic boundary conditions prescribed at the edges of the plate, and the fluid layer height. It has been found that a violation of the smoothness of the obtained relationships and diagrams of stability is caused by a change in the flutter mode, or change of the type of loss of stability.

Détail

Keywords: aeroelasticity, supersonic gas flow, potential flow of compressible fluid, rectangular plate, finite-element method, stability, flutter.

Bibliographie

[1] Dowell E. H., Aeroelasticity of Plates and Shells, Mechanics: Dynamical Systems, 1, Springer, Netherlands, 1975, xiii+139 pp.

[2] Ilyushin A. A., “The law of plane sections in aerodynamics of the high supersonic velocities”, Prikl. Mat. Mekh, 20:6 (1956), 733–755 (In Russian)

[3] Novichkov Yu. N., “Flutter of plates and shells”, Itogi nauki i tekhniki [Advances in Science and Technology], Mechanics of Deformable Solids, 11, VINITI, Moscow, 1978, 67–122 (In Russian) | MR

[4] Dowell E. H., Voss H. M., “Theoretical and experimental panel flutter studies in the Mach number range 1.0 to 5.0”, AIAA J., 3:12 (1965), 2292–2304 | DOI

[5] Dixon S. C., Comparison of panel flutter results from approximate aerodynamic theory with results from exact inviscid theory and experiment, NASA TN D-3649; Technical Report no. 19660029122, 1966 https://ntrs.nasa.gov/search.jsp?R=19660029122

[6] Vedeneev V. V., Guvernyuk S. V., Zubkov A. F., Kolotnikov M. E., “Experimental investigation of single-mode panel flutter in supersonic gas flow”, Fluid Dyn., 45:2 (2010), 312–324 | DOI | MR

[7] Bismarck-Nasr M. N., “Finite element analysis of aeroelasticity of plates and shells”, Appl. Mech. Rev., 45:12 (1992), 461–482 | DOI

[8] Païdoussis M. P., Fluid-structure Interactions: Slender Structures and Axial Flow, 2nd edition, v. 2, Academic Press, London, 2014, 923 pp. | DOI

[9] Kornecki A., Dowell E. H., O'Brien J., “On the aeroelastic instability of two-dimensional panels in uniform incompressible flow”, Journal of Sound and Vibration, 47:2 (1976), 163–178 | DOI | Zbl

[10] Dowell E. H., “Nonlinear oscillations of a fluttering plate”, AIAA J., 4:7 (1966), 1267–1275 | DOI

[11] Dowell E. H., “Nonlinear oscillations of a fluttering plate. II”, AIAA J., 5:10 (1967), 1856–1862 | DOI

[12] Mei C., “A finite-element approach for nonlinear panel flutter”, AIAA J., 15:8 (1977), 1107–1110 | DOI

[13] Yang T. Y., Han A. D., “Nonlinear panel flutter using high-order triangular finite elements”, AIAA J., 21:10 (1983), 1453–1461 | DOI | Zbl

[14] Haddadpour H., Navazi H. M., Shadmehri F., “Nonlinear oscillations of a fluttering functionally graded plate”, Compos. Struct., 79:2 (2007), 242–250 | DOI

[15] Prakash T., Ganapathi M., “Supersonic flutter characteristics of functionally graded flat panels including thermal effects”, Compos. Struct., 72:1 (2006), 10–18 | DOI

[16] Rossettos J. N., Tong P., “Finite-element analysis of vibration and flutter of cantilever anisotropic plates”, J. Appl. Mech., 41:4 (1974), 1075–1080 | DOI | Zbl

[17] Bochkarev S. A., Lekomtsev S. V., Matveenko V. P., “Hydroelastic stability of rectangular plate interacting with a layer of ideal flowing fluid”, Fluid Dyn., 51:6 (2016) | DOI | Zbl

[18] Bochkarev S. A., Lekomtsev S. V., “Numerical investigation of the effect of boundary conditions on hydroelastic stability of two parallel plates interacting with a layer of ideal flowing fluid”, Comput. Continuum Mech., 8:4 (2015), 423–432 (In Russian) | DOI

[19] Kerboua Y., Lakis A. A., Thomas M., Marcouiller L., “Modeling of plates subjected to a flowing fluid under various boundary conditions”, Eng. Appl. Comp. Fluid Mech., 2:4 (2008), 525–539 | DOI

[20] Noorian M., Haddadpour H., Firouz-Abadi R., “Investigation of panel flutter under the effect of liquid sloshing”, J. Aerosp. Eng., 28:2 (2015), 04014059 | DOI

[21] Bochkarev S. A., Lekomtsev S. V., “An aeroelastic stability of the circular cylindrical shells containing a flowing fluid”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 19:4 (2015), 750–767 (In Russian) | DOI

[22] Vol'mir A. S., Nelineinaia dinamika plastin i obolochek [Nonlinear Dynamics of Plates and Shells], Nauka, Moscow, 1972, 432 pp. (In Russian) | MR

[23] Vol'mir A. S., Obolochki v potoke zhidkosti i gaza. Zadachi gidrouprugosti [Shells in Fluid and Gas Flow. Problems of Hydroelasticity], Nauka, Moscow, 1979, 320 pp. (In Russian)

[24] Lehoucq R. B., Sorensen D. C., “Deflation techniques for an implicitly restarted Arnoldi iteration”, SIAM J. Matrix Anal. Appl., 17:4 (1996), 789–821 | DOI | MR | Zbl

[25] Olson M. D., “Some flutter solutions using finite elements”, AIAA J., 8:4 (1970), 747–752 | DOI | MR