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The Flow of A Falling Ellipse: Numerical Method and Classification

Published online by Cambridge University Press:  20 August 2015

R.-J. Wu  and
S.-Y. Lin
R.-J. Wu
Affiliation:
Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan, Taiwan
S.-Y. Lin*
Affiliation:
Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan, Taiwan
*
* Corresponding author ([email protected])
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Abstract

A modified direct-forcing immersed-boundary (IB) pressure correction method is developed to simulate the flows of a falling ellipse. The pressure correct method is used to solve the solutions of the two dimensional Navier-Stokes equations and a direct-forcing IB method is used to handle the interaction between the flow and falling ellipse. For a fixed aspect ratio of an ellipse, the types of the behavior of the falling ellipse can be classified as three pure motions: Steady falling, fluttering, tumbling, and two transition motions: Chaos, transition between steady falling and fluttering. Based on two dimensionless parameters, Reynolds number and the dimensionless moment of inertia, a Reynolds number-inertia moment phase diagram is established. The behaviors and characters of five falling regimes are described in detailed.

Keywords

Falling ellipsePressure correction methodFlutteringTumbling
Type
Research Article
Information
Journal of Mechanics , Volume 31 , Issue 6 , December 2015 , pp. 771 - 782
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2015 

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References

1.Andersen, A., Pesavento, U. and Wang, Z. J., “Unsteady Aerodynamics of Fluttering and Tumbling Plates,” Journal of Fluid Mechanics, 541, pp. 65–90 (2005).CrossRefGoogle Scholar
2.Andersen, A., Pesavento, U. and Wang, Z. J., “Analysis of Transitions Between Fluttering, Tumbling and Steady Descent of Falling Cards,” Journal of Fluid Mechanics, 541, pp. 91–104 (2005).Google Scholar
3.Jin, C. and Xu, K., “Numerical Study of the Unsteady Aerodynamics of the Freely Falling Plates,” Communications in Computational Physics, 3, pp. 834–851 (2008).Google Scholar
4.Kolomenskiy, D.amd Schneider, K., “Numerical Simulations of Falling Leaves Using a Pseudo–Spectral Method with Volume Penalization,” Theoretical and Computational Fluid Dynamics, 24, pp.Google Scholar
5.Smith, E. H., “Autorotating Wings: An Experimental Investigation,” Journal of Fluid Mechanics, 50, pp. 513–534 (1971).CrossRefGoogle Scholar
6.Tanabe, Y. and Kaneko, K., “Behavior of a Falling Paper,” Physical Review Letters, 73, pp. 1372–1421 (1994).CrossRefGoogle ScholarPubMed
7.Belmonte, A., Eisenberg, H. and Moses, E., “From Flutter to Tumble: Inertial Drag and Frounde Similarity in Falling Paper,” Physical Review Letters, 81, pp. 345–348 (1998).CrossRefGoogle Scholar
8.Mahadevan, L., Ryu, W. S.and Samuel, A. D. T., “Tumbling Cards,” Physics Fluids, 11, 111070–6631/11(1) (1999).Google Scholar
9.Pesavento, U. and Wang, Z. J., “Falling Paper: Navier–Stokes Solutions, Model of Fluid Forces, and Center of Mass Elevation,” Physical Review Letters, 93, 144501 (2004).Google Scholar
10.Haeri, S. and Shrimpton, J. S., “On the Application of Immersed Boundary, Fictitious Domain and Body–Conformal Mesh Methods to Many Particle Multiphase Flows,” International Journal of Multiphase Flow, 40, pp. 38–55 (2012).Google Scholar
11.Hu, H., “Direct Simulation of Flows of Solid–Liquid Mixtures,” International Journal of Multiphase Flow, 22, pp. 332–335 (1996).Google Scholar
12.Johnson, A. A.and Tezduyar, T. E., “Simulation of Multiple Spheres Falling in a Liquid–Filled Tube,” Computer Methods in Applied Mechanics and Engineering, 134, pp. 351–373 (1996).Google Scholar
13.Johnson, A. A.and Tezduyar, T. E., “Advanced Mesh Generation and Update Methods for 3D Flow Simulations,” Computational Mechanics, 23, pp. 130–143 (1999).CrossRefGoogle Scholar
14.Hu, H., Joseph, D. and Crochet, M., “Direct Simulation of Fluid Particle Motions,” Theoretical and Computational Fluid Dynamics, 3, pp. 285–306 (1992).Google Scholar
15.Gan, H., Chang, J., Feng, J. and Hu, H., “Direct Numerical Simulation of the Sedimentation of Solid Particles with Thermal Convection,” Journal of Fluid Mechanics, 481, pp. 385–411 (2003).CrossRefGoogle Scholar
16.Johnson, A. and Tezduyar, T., “Models for 3D Computations of Fluid–Particle Interactions in Spatially Periodic Flows,” Computer Methods in Applied Mechanics and Engineering, 190, pp. 3201–3221 (2001).CrossRefGoogle Scholar
17.Wan, D. and Turek, S., “Direct Numerical Simulation of Particulate Flow via Multigrid FEM Techniques and the Fictitious Boundary Method,” International Journal for Numerical Methods in Fluids, 51, pp. 531–566 (2006).CrossRefGoogle Scholar
18.Wan, D. and Turek, S., “Fictitious Boundary and Moving Mesh Methods for the Numerical Simulation of Rigid Particulate Flows,” Journal of Computational Physics, 222, pp. 28–56 (2007).CrossRefGoogle Scholar
19.Peskin, C. S., “Numerical Analysis of Blood Flow in the Heart,” Journal of Computational Physics, 25, pp. 220–252 (1977).CrossRefGoogle Scholar
20.Höfler, K. and Schwarzer, S., “Navier–Stokes Simulation with Constraint Forces: Finite–Difference Method for Particle–Laden Flows and Complex Geometries,” Physical Review E, 61, pp. 7146–7160 (2000).CrossRefGoogle ScholarPubMed
21.Feng, Z. G.and Michaelides, E. E., “The Immersed Boundary–Lattice Boltzmann Method for Solving Fluid–Particles Interaction Problems,” Journal of Computational Physics, 195, pp. 602–628 (2004).Google Scholar
22.Lai, M. C.and Peskin, C., “An Immersed Boundary Method with Formal Second–Order Accuracy and Reduced Numerical Viscosity,” Journal of Computational Physics, 160, pp. 705–719 (2000).CrossRefGoogle Scholar
23.Lee, C., “Stability Characteristics of the Virtual Boundary Method in Three–Dimensional Applications,” Journal of Computational Physics, 184, pp. 559–591 (2003).Google Scholar
24.Fadlun, E. A., Verizicco, R., Orlandi, P. and Mohd–Yusof, J., “Combined Immersed–Boundary Finite–Difference Methods for Three–Dimensional Complex Flow Simulations,” Journal of Computational Physics, 161, pp. 35–60 (2000).Google Scholar
25.Feng, Z. G.and Michaelides, E. E., “Proteus: A Direct Forcing Method in the Simulations of Particulate Flows,” Journal of Computational Physics, 202, pp. 20–51 (2005).CrossRefGoogle Scholar
26.Uhlmann, M., “An Immersed Boundary Method with Direct Forcing for the Simulation of Particulate Flows,” Journal of Computational Physics, 209, pp. 448–476 (2005).Google Scholar
27.Lin, S. Y., Chin, Y. H., Hu, J. J.and Chen, Y. C., “A Pressure Correction Method for Fluid–Particle Interaction Flow: Direct Forcing Method and Sedimentation Flow,” International Journal for Numerical Methods in Fluids, 67, pp. 1171–1798 (2011).Google Scholar
28.Dennis, S. C. R. and Chang, G. Z., “Numerical Integration of the Navier–Stokes Equations for Steady Two–Dimensional Flow,” Physics of Fluids, 12, pp. 88–93 (1969).Google Scholar
29.Lugt, H. J.and Haussling, H. J., “Laminar Flows Past Elliptic Cylinders at Various Angles of Attack,” Naval Ship Research and Development Center, 3748 (1972).Google Scholar
30.Mittal, R., Seshadri, V., Udaykumar, H. S., “Flutter, Tumble and Vortex Induced Autorotation,” Theoretical and Computational Fluid Dynamics, 17, pp. 165–170 (2004).Google Scholar