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Three-dimensional flow instability in a lid-driven isosceles triangular cavity

Published online by Cambridge University Press:  22 March 2011

L. M. GONZÁLEZ
Affiliation:
School of Naval Engineering, Universidad Politécnica de Madrid, Pza. Cardenal Cisneros 3, E-28040 Madrid, Spain
M. AHMED
Affiliation:
Institute of Fluid Mechanics and Heat Transfer, Vienna University of Technology, Resselgasse 3/1/2, A-1040 Vienna, Austria
J. KÜHNEN
Affiliation:
Institute of Fluid Mechanics and Heat Transfer, Vienna University of Technology, Resselgasse 3/1/2, A-1040 Vienna, Austria
H. C. KUHLMANN
Affiliation:
Institute of Fluid Mechanics and Heat Transfer, Vienna University of Technology, Resselgasse 3/1/2, A-1040 Vienna, Austria
V. THEOFILIS
Affiliation:
School of Aeronautics, Universidad Politécnica de Madrid, Pza. Cardenal Cisneros 3, E-28040 Madrid, Spain
Corresponding

Abstract

Linear three-dimensional modal instability of steady laminar two-dimensional states developing in a lid-driven cavity of isosceles triangular cross-section is investigated theoretically and experimentally for the case in which the equal sides form a rectangular corner. An asymmetric steady two-dimensional motion is driven by the steady motion of one of the equal sides. If the side moves away from the rectangular corner, a stationary three-dimensional instability is found. If the motion is directed towards the corner, the instability is oscillatory. The respective critical Reynolds numbers are identified both theoretically and experimentally. The neutral curves pertinent to the two configurations and the properties of the respective leading eigenmodes are documented and analogies to instabilities in rectangular lid-driven cavities are discussed.

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Copyright © Cambridge University Press 2011

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References

Ahusborde, E. & Glockner, S. 2010 A 2d block-structured mesh partitioner for accurate flow simulations on non-rectangular geometries. Comput. Fluids, doi:10.1016/j.compfluid.2010.07.009.Google Scholar
Aidun, C. K., Triantafillopoulos, N. G. & Benson, J. D. 1991 Global stability of a lid-driven cavity with throughflow: Flow visualization studies. Phys. Fluids A 3, 20812091.CrossRefGoogle Scholar
Albensoeder, S. & Kuhlmann, H. C. 2002 a Linear stability of rectangular cavity flows driven by anti-parallel motion of two facing walls. J. Fluid Mech. 458, 153180.CrossRefGoogle Scholar
Albensoeder, S. & Kuhlmann, H. C. 2002 b Three-dimensional instability of two counter-rotating vortices in a rectangular cavity driven by parallel wall motion. Eur. J. Mech. B/Fluids 21 (3), 307316.CrossRefGoogle Scholar
Albensoeder, S. & Kuhlmann, H. C. 2003 Stability balloon for the double-lid-driven cavity flow. Phys. Fluids 15, 24532457.CrossRefGoogle Scholar
Albensoeder, S. & Kuhlmann, H. C. 2006 Nonlinear three-dimensional flow in the lid-driven square cavity. J. Fluid Mech. 569, 465480.CrossRefGoogle Scholar
Albensoeder, S., Kuhlmann, H. C. & Rath, H. J. 2001 a Multiplicity of steady two-dimensional flows in two-sided lid-driven cavities. Theor. Comput. Fluid Dyn. 14, 223241.CrossRefGoogle Scholar
Albensoeder, S., Kuhlmann, H. C. & Rath, H. J. 2001 b Three-dimensional centrifugal-flow instabilities in the lid-driven-cavity problem. Phys. Fluids 13, 121135.CrossRefGoogle Scholar
Batchelor, G. K. 1956 On steady laminar flow with closed streamlines at large Reynolds number. J. Fluid Mech. 1, 177190.CrossRefGoogle Scholar
Boppana, V. B. L. & Gajjar, J. S. B. 2010 Global flow instability in a lid-driven cavity. Intl J. Numer. Meth. Fluids 62 (8), 827853.Google Scholar
Brezillon, A., Girault, G. & Cadou, J. M. 2010 A numerical algorithm coupling a bifurcating indicator and a direct method for the computation of Hopf bifurcation points in fluid mechanics. Comput. Fluids 39 (7), 12261240.CrossRefGoogle Scholar
Burggraf, O. R. 1966 Analytical and numerical studies of the structure of steady separated flows. J. Fluid Mech. 24, 113151.CrossRefGoogle Scholar
Darr, J. H. & Vanka, S. P. 1991 Separated flow in a driven trapezoidal cavity. Phys. Fluids A 3, 385392.CrossRefGoogle Scholar
Deville, M., , T.-H & Morchoisne, Y. 1992 Numerical simulation of 3-D incompressible unsteady viscous laminar flows. Notes on Numerical Fluid Mechanics, vol. 36, Vieweg, Braunschweig.Google Scholar
Ding, Y. & Kawahara, M. 1998 Linear stability of incompressible flow using a mixed finite element method. J. Comput. Phys. 139, 243273.CrossRefGoogle Scholar
Erturk, E. & Gokcol, O. 2007 Fine grid numerical solutions of triangular cavity flow. Eur. Phys. J. Appl. Phys. 38, 97105.CrossRefGoogle Scholar
Feldman, Y. & Gelfgat, A. Y. 2010 Oscillatory instability of a 3d lid-driven flow in a cube. Phys. Fluids 22, 093602.CrossRefGoogle Scholar
Freitas, C. J., Street, R. L., Findikakis, A. N. & Koseff, J. R. 1985 Numerical simulation of three-dimensional flow in a cavity. Intl J. Numer. Meth. Fluids 5, 561575.CrossRefGoogle Scholar
Gaskell, P. H., Thompson, H. M. & Savage, M. D. 1999 A finite element analysis of steady viscous flow in triangular cavities. Proc. Inst. Mech. Engng C: J. Mech. Engng Sci., 213, 263276.CrossRefGoogle Scholar
Ghia, U., Ghia, K. N. & Shin, C. T. 1982 High-Re solutions for incompressible flow using the Navier–Stokes equations and a multigrid method. J. Comput. Phys. 48, 387411.CrossRefGoogle Scholar
Giannetti, F., Luchini, P. & Marino, L. 2009 Linear stability analysis of three-dimensional lid-driven cavity flow. In Atti del XIX Congresso AIMETA di Meccanica Teorica e Applicata, 1417 Sep. 2009. Ancona, Italy.Google Scholar
Giannetti, F., Luchini, P. & Marino, L. 2010 Characterization of the three-dimensional instability in a lid-driven cavity by an adjoint based analysis. In Proceedings of the Seventh IUTAM Symposium on Laminar-Turbulent Transition, Stockholm, Sweden, pp. 165170.CrossRefGoogle Scholar
González, L. M. & Bermejo, R. 2005 A semi-langrangian level set method for incompressible Navier–Stokes equations with free surface. Intl J. Numer. Meth. Fluids 49, 11111146.Google Scholar
González, L. M., Theofilis, V. & Gómez-Blanco, R. 2007 Finite-element numerical methods for viscous incompressible BiGlobal linear instability analysis on unstructured meshes. AIAA J. 45, 840854.CrossRefGoogle Scholar
Guj, G. & Stella, F. 1993 A vorticity–velocity method for the numerical solution of 3-d incompressible flows. J. Comput. Phys. 106, 286298.CrossRefGoogle Scholar
Hill, D. C. 1992 A theoretical approach for the restabilization of wakes. In 30th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, AIAA Paper 92-0067.Google Scholar
Hill, D. C. 1995 Adjoint systems and their role in the receptivity problem for boundary layers. J. Fluid Mech. 292, 183204.CrossRefGoogle Scholar
Hinatsu, M. & Ferziger, J. H. 1991 Numerical computation of unsteady incompressible flow in complex geometry using a composite multigrid technique. Intl J. Numer. Meth. Fluids 13 (8), 971997.CrossRefGoogle Scholar
Humphrey, J. A. C., Cushner, J., Al-Shannag, M., Herrero, J. & Giralt, F. 2003 Shear-driven flow in a toroid of square cross section. ASME J. Fluids Engng 125, 130137.CrossRefGoogle Scholar
Jyotsna, R. & Vanka, S. P. 1995 Multigrid calculation of steady, viscous flow in a triangular cavity. J. Comput. Phys. 122, 107117.CrossRefGoogle Scholar
Kawaguti, M. 1961 Numerical solution of the Navier–Stokes equations for the flow in a two-dimensional cavity. J. Phys. Soc. Japan 16, 23072315.CrossRefGoogle Scholar
Koseff, J. R. & Street, R. L. 1984 a The lid-driven cavity flow: A synthesis of qualitative and quantitative observations. J. Fluids Engng 106 (4), 390398.CrossRefGoogle Scholar
Koseff, J. R. & Street, R. L. 1984 b On end wall effects in a lid-driven cavity flow. J. Fluids Engng 106 (4), 385389.CrossRefGoogle Scholar
Koseff, J. R., Street, R. L., Gresho, P. M., Upson, C. D., Humphrey, J. A. C. & To, W. M. 1983 Three-dimensional lid-driven cavity flow: experiment and simulation. In International Conference on Numerical Methods in Laminar and Turbulent Flow, 8 Aug., Seattle, WA.Google Scholar
Ku, H. C., Hirsch, R. S. & Taylor, T. D. 1987 A pseudospectral method for solution of the three-dimensional incompressible Navier–Stokes equations. J. Comput. Phys. 70 (2), 439462.CrossRefGoogle Scholar
Kuhlmann, H. C., Wanschura, M. & Rath, H. J. 1997 Flow in two-sided lid-driven cavities: non-uniqueness, instabilities, and cellular structures. J. Fluid Mech. 336, 267299.CrossRefGoogle Scholar
Leriche, E. 2006 Direct numerical simulation in a lid-driven cubical cavity at high Reynolds number by a Chebyshev spectral method. J. Sci. Comput. 27 (1–3), 335345.CrossRefGoogle Scholar
Leriche, E. & Gavrilakis, S. 2000 Direct numerical simulation of the flow in a lid-driven cubical cavity. Phys. Fluids 12 (6), 13631376.CrossRefGoogle Scholar
Leriche, E., Gavrilakis, S. & Deville, M. O. 1998 Direct simulation of the lid-driven cavity flow with Chebyshev polynomials. In Proceedings of the 4th European Computational Fluid Dynamics Conference (ed. Papailiou, K. D.), vol. 1(1), pp. 220225. Athens, Greece.Google Scholar
Li, M. & Tang, T. 1996 Steady viscous flow in a triangular cavity by efficient numerical techniques. Comput. Maths Applics. 31, 5565.CrossRefGoogle Scholar
Luchini, P. 1991 A deferred-correction multigrid algorithm based on a new smoother for the Navier–Stokes equations. J. Comput. Phys. 92 (2), 349368.CrossRefGoogle Scholar
McQuain, W. D., Ribbens, C. J., Wang, C.-Y. & Watson, L. T. 1994 Steady viscous flow in a trapezoidal cavity. Comput. Fluids 23, 613626.CrossRefGoogle Scholar
Moffat, H. K. 1963 Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18, 118.CrossRefGoogle Scholar
Nithiarasu, P. & Liu, C.-B. 2005 Steady and unsteady incompressible flow in a double driven cavity using the artificial compressibility (AC)-based characteristic-based split (CBS) scheme. Intl J. Numer. Meth. Engng 63, 380397.CrossRefGoogle Scholar
Oosterlee, C. W., Wesseling, P., Segal, A. & Brakkee, E. 1993 Benchmark solutions for the incompressible Navier–Stokes equations in general coordinates on staggered grids. Intl J. Numer. Meth. Fluids 17, 301321.CrossRefGoogle Scholar
Pan, F. & Acrivos, A. 1967 Steady flows in rectangular cavities. J. Fluid Mech. 28, 643655.CrossRefGoogle Scholar
Perng, C. Y. & Street, R. L. 1991 A coupled multigrid-domain-splitting technique for simulating incompressible flows in geometrically complex domains. Intl J. Numer. Meth. Fluids 13 (3), 269286.CrossRefGoogle Scholar
Poliashenko, M. & Aidun, C. K. 1995 A direct method for computation of simple bifurcations. J. Comput. Phys. 121 (2), 246260.CrossRefGoogle Scholar
Prasad, A. K. & Koseff, J. R. 1989 Reynolds number and end-wall effects on a lid-driven cavity flow. Phys. Fluids A 1, 208218.CrossRefGoogle Scholar
Rhee, H. S., Koseff, J. R. & Street, R. L. 1984 Flow visualization of a recirculating flow by rheoscopic liquid and liquid crystal techniques. Exp. Fluids 2 (2), 5764.CrossRefGoogle Scholar
Ribbens, C. J., Wang, C.-Y., Watson, L. T. & Alexander, K. A. 1991 Vorticity induced by a moving elliptic belt. Comput. Fluids 20, 111119.CrossRefGoogle Scholar
Ribbens, C. J., Watson, L. T. & Wang, C.-Y. 1994 Steady viscous flow in a triangular cavity. J. Comput. Phys. 112, 173181.CrossRefGoogle Scholar
Saad, Y. 1980 Variations of Arnoldi's method for computing eigenelements of large unsymmetric matrices. Linear Algebr. Applics. 34, 269295.CrossRefGoogle Scholar
Schreiber, R. & Keller, H.B. 1983 Driven cavity flows by efficient numerical techniques. J. Comput. Phys. 49, 310333.CrossRefGoogle Scholar
Shankar, P. N. & Deshpande, M. D. 2000 Fluid mechanics in the driven cavity. Annu. Rev. Fluid Mech. 32, 93136.CrossRefGoogle Scholar
Siegmann-Hegerfeld, T., Albensoeder, S. & Kuhlmann, H. C. 2007 Experiments on the transition to three-dimensional flow structures in two-sided lid-driven cavities. Proc. Appl. Math. Mech. 7, 3050001 (12).CrossRefGoogle Scholar
Siegmann-Hegerfeld, T., Albensoeder, S. & Kuhlmann, H. C. 2008 Two- and three-dimensional flows in nearly rectangular cavities driven by collinear motion of two facing walls. Exp. Fluids 45, 781796.CrossRefGoogle Scholar
Simuni, L. M. 1965 Numerical solution of the problem of the motion of a fluid in a rectangular hole. J. Appl. Mech. Tech. Phys. 6, 106108.Google Scholar
Theofilis, V. 2000 Globally unstable basic flows in open cavities. In Sixth AIAA Aeroacoustics Conference and Exhibit, Lahaina, HI, AIAA Paper 2000-1965.Google Scholar
Theofilis, V. 2003 Advances in global linear instability analysis of nonparallel and three-dimensional flows. Prog. Aero. Sci. 39, 249315.CrossRefGoogle Scholar
Theofilis, V. 2011 Global linear instability. Annu. Rev. Fluid Mech. 43, 319352.CrossRefGoogle Scholar
Verstappen, R. W. C. P. & Veldman, A. E. P. 1994 Direct numerical simulation of a 3D turbulent flow in a driven cavity at Re=10 000. In Proceedings of Second Fluids Dynamics Conference, ECCOMAS, p. 558. Stuttgart, Germany.Google Scholar
de Vicente, J., Rodriguez, D., Theofilis, V. & Valero, E. 2010 a On high-Re numerical solutions in spanwise-periodic lid-driven cavity flows with complex cross-sectional profiles, doi:10.1016/j.compfluid.2010.09.033.Google Scholar
de Vicente, J., Theofilis, V. & Valero, E. 2010 b Multi-domain spectral collocation for BiGlobal instability analysis on non-conforming Cartesian subdomains. Euro. J. Mech./B Fluids (submitted).Google Scholar
Vynnycky, M. & Kimura, S. 1994 An investigation of recirculating flow in a driven cavity. Phys. Fluids 6, 36103620.CrossRefGoogle Scholar
Zhou, Y. C., Patnaik, B. S. V., Wan, D. C. & Wei, G. W. 2003 DSC solution for flow in a staggered double lid driven cavity. Intl J. Numer. Meth. Engng 57, 211234.CrossRefGoogle Scholar

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