Stokes flow in a rectangular cavity with two moving lids (with equal speed but in opposite directions) and aspect ratio A (height to width) is considered. An analytic solution for the streamfunction, ψ, expressed as an infinite series of Papkovich–Fadle eigenfunctions is used to reveal changes in flow structures as A is varied. Reducing A from A=0.9 produces a sequence of flow transformations at which a saddle stagnation point changes to a centre (or vice versa) with the generation of two additional stagnation points. To obtain the local flow topology as A→0, we expand the velocity field about the centre of the cavity and then use topological methods. Expansion coefficients depend on the cavity aspect ratio which is considered as a separation parameter. The normal-form transformations result in a much simplified system of differential equations for the streamlines encapsulating all features of the original system. Using the simplified system, streamline patterns and their bifurcations are obtained, as A→0.
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Bakker, P.G. (1989). Bifurcation in Flow Patterns. Thesis, Technical University of Delft.
Brøns, M., and Hartnack, J.N. (1999). Streamline topologies near simple degenerate critical points in two-dimensional flow away from boundaries. Phys. Fluids, 11, 314–324.
Burggraf, O. (1966). Analytical and numerical studies of the structure of steady separated flows. J. Fluid Mech., 24, 113–151.
Dallmann, U. (1988). Three-dimensional vortex structures and vorticity topology. Fluid Dyn. Res., 3, 183–192.
Dean, W.R. (1950). Note on the motion of liquid near a position of separation. Proc. Cambridge Philos. Soc., 46, 293–306.
Gaskell, P.H., Lau, A.K.C., and Wright, N.G. (1988). Comparison of two solution strategies for use with higher-order discretization schemes in fluid flow simulation. Int. J. Numer. Methods. Eng., 8, 1203–1215.
Gaskell, P.H., Savage, M.D., and Wilson, M. (1997). Flow structure in a half-filled annulus between rotating co-axial cylinders. J. Fluid Mech., 337, 263–282.
Gaskell, P.H., Gürcan, F., Savage, M.D., and Thompson, H.M. (1998). Stokes flow in a double-lid-driven cavity with free surface side-walls. J. Mech. Eng. Sci. C, 212(5), 387–403.
Gatski, T.B., Grosch, C.E., and Rose, M.E. (1982). A numerical study of the two dimensional Navier–Stokes equation in vorticity–velocity variables. J. Comput. Phys. 48, 1–22.
Goldstein, H. (1950). Classical Mechanics. Addison-Wesley, Reading, MA.
Hartnack, J.N. (1999). Streamlines topologies near a fixed wall using normal forms. Acta Mech., 136, 55–75.
Joseph, D.D., and Sturges, L. (1978). The convergence of biorthogonal series for biharmonic and Stokes flow edge problems: Part II. SIAM. J. Appl. Math., 34, 7–27.
Legendre, R. (1956). Séparation de I’écoulement laminaire tridimensionel. Reah. Aerosp., 54(3), 3–8.
Oswatitsch, K. (1958). Die Ablosungsbedingung von Grenzschichten. In IUTAM Symposium on Boundary Layer Resarch, 357–364. Springer-Verlag, Berlin.
Pan, F., and Acrivos, A. (1967). Steady flows in rectangular cavities. J. Fluid Mech., 28, 643–655.
Perry, A.E., and Chong, M.S. (1987). A description of eddying motions and flow pattens using critical–point concepts. Annu. Rev. Fluid Mech., 19, 125–137.
Robbins, C.I., and Smith, R.C.T. (1948). A table of roots of sin z = -z. Philos. Mag., 7, 39, 1005.
Shankar, P.N. (1993). The eddy structure in Stokes flow in a cavity. J. Fluid Mech., 250, 371–383.
Shankar, P.N., and Deshpande, M.N. (2000). Fluid mechanics in the driven cavity. Annu. Rev. Fluid Mech., 32, 93–136.
Shen, C., and Floryan, J.M. (1985). Low Reynolds number flow over cavities., Phys. Fluids, 28(11), 3191–3202.
Smith, R.C.T. (1952). The bending of a semi-infinite strip. Aust. J. Sci. Res., 5, 227–237.
Sturges, L. (1986). Stokes flow in a two-dimensional cavity with moving end walls. Phys. Fluids, 29(5), 1731–1734.
Tobak, M., and Peake, D.J. (1982). Topology of three-dimensional separated flows. Annu. Rev. Fluid Mech., 14, 61–85.
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Gürcan, F. Streamline Topologies in Stokes Flow Within Lid-Driven Cavities. Theoret Comput Fluid Dynamics 17, 19–30 (2003). https://doi.org/10.1007/s00162-003-0095-z
- Aspect Ratio
- Velocity Field
- Flow Structure
- Point Change
- Original System