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The Lid-Driven Cavity

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Computational Modelling of Bifurcations and Instabilities in Fluid Dynamics

Part of the book series: Computational Methods in Applied Sciences ((COMPUTMETHODS,volume 50))

Abstract

The lid-driven cavity is an important fluid mechanical system serving as a benchmark for testing numerical methods and for studying fundamental aspects of incompressible flows in confined volumes which are driven by the tangential motion of a bounding wall. A comprehensive review is provided of lid-driven cavity flows focusing on the evolution of the flow as the Reynolds number is increased. Understanding the flow physics requires to consider pure two-dimensional flows, flows which are periodic in one space direction as well as the full three-dimensional flow. The topics treated range from the characteristic singularities resulting from the discontinuous boundary conditions over flow instabilities and their numerical treatment to the transition to chaos in a fully confined cubical cavity. In addition, the streamline topology of two-dimensional time-dependent and of steady three-dimensional flows are covered, as well as turbulent flow in a square and in a fully confined lid-driven cube. Finally, an overview on various extensions of the lid-driven cavity is given.

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Notes

  1. 1.

    Here the convective scaling with characteristic velocity U is used to facilitate the mathematical analysis.

  2. 2.

    The deep penetration of the wall jet observed experimentally by Pan and Acrivos [254], which is in contradiction with the results for pure two-dimensional single-lid-driven flows (Sect. 5.1), might have been caused by the strong geometric confinement in z of the flow in their experiments with \(L_z=L_y\).

  3. 3.

    Like the elliptic instability, the quadripolar instability is due to a Kelvin-wave resonance, communicated by a quadripolar strain field [8, 102]. Since the resonance condition in the ideal case of a columnar vortex (see e.g. Chandrasekhar [61]) can only be satisfied for asymptotically large wave number k, the critical wave number in the lid-driven cavity flow is quite large: \(k_c\approx 15\) (see Fig. 20).

  4. 4.

    A dynamical system is called hyperbolic in a linear sense when the Jacobian determining the local linearization of the flow admits non-imaginary eigenvalues.

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Acknowledgements

We are very grateful to S. Albensoeder, F. Auteri, O. Botella, R. Bouffanais, C.-H. Bruneau, G. Courbebaisse, J. R. Koseff, E. Leriche, J.-C. Loiseau, J. M. Lopez, H. K. Moffatt, J. M. Ottino, A. Povitsky, W. W. Schultz, J. F. Scott and T. W. H. Sheu, who kindly allowed the reproduction of their figures which have been published earlier.

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    Hendrik C. Kuhlmann & Francesco Romanò

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Kuhlmann, H.C., Romanò, F. (2019). The Lid-Driven Cavity. In: Gelfgat, A. (eds) Computational Modelling of Bifurcations and Instabilities in Fluid Dynamics. Computational Methods in Applied Sciences, vol 50. Springer, Cham. https://doi.org/10.1007/978-3-319-91494-7_8

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