Over the years the single lid-driven cavity flow has been used as a benchmark problem to test the performance of numerical schemes and algorithms for incompressible flows. The problem has attracted researchers because it contains a wide variety of interesting phenomenon in the simplest of geometric settings. The single lid-driven cavity flow was extended to two- and four-sided cavity flows by various investigators [1, 2, 3, 4, 5, 6, 7], who observed that a plethora of vortex patterns can be generated with different aspect ratios and directions of motion of the walls.

It is well known that many nonlinear problems exhibit multiple steady solutions even though the governing equations and boundary conditions remain the same. As the governing equations for fluid flow are nonlinear in nature, the possibility of multiple solutions exists. Many researchers have found multiple solutions for parallel wall motion of facing walls for both rectangular and square cavities, and for antiparallel wall motion for rectangular cavities [1, 2, 3, 4, 5, 6, 7]. Albensoeder et al. [2] were among the first to investigate the nonlinear regime and find multiple 2D steady states in rectangular two-sided lid-driven cavities. They have found upto five different flow states for both parallel and antiparallel motion of facing walls. Very recently Lemée et al. [4] addressed the issue of multiple solutions in square cavity with parallel motion of facing walls and found out a critical Reynolds number above which multiple solutions exist, which is consistent with [2]. Similar investigations have been carried out for two-sided cavities with motion of non-facing walls and for four-sided cavities [1, 5]. All these existing investigations have been carried out using non-compact schemes. It can be observed from these investigations, that the additional solutions, if they exist, always exist in pairs.

In this work, we re-examine these solutions using a higher-order compact (HOC) scheme of spatially fourth- and temporal second-order accuracy. This scheme was developed by Kalita et. al [8]  by differentiating the governing equation to obtain compact approximations for the leading truncation error terms. Grid independent results are carefully computed so that the results can be used as means to test other schemes and algorithms. For the two-sided rectangular cavity, computations are carried out at a fixed Reynolds number (Re) of 600 at various aspect ratios (A) for parallel motion of facing walls. It is seen that at Re=600 multiple solutions exist only above a critical aspect ratio of 0.556. This value is very close to the value 0.559 reported in [2]. Computations are also carried out for square cavities having antiparallel and parallel motion of facing walls at various Re's. For parallel wall motion in a square cavity, a threshold value of Re=983.5 is observed below which only stable symmetric solutions exist. This value is in good agreement with previously reported values in [2,4]. Fig. 1(a) shows the geometry for two-sided parallel cavity with motion of facing walls. The multiple solutions can be seen for this configuration in Fig. 2 at Re=4000. For antiparallel motion of facing walls in a square cavity, till very recently, existence of multiple steady solutions was not experienced. In a recent communication [9], using the same HOC scheme as the one used here, we demonstrate that existence. This shows the accuracy and effectiveness of the present scheme, which we use here to compute multiple solutions for the motion of non–facing walls in two- and four-sided configurations as well. Fig. 3 shows the multiple solutions observed at Re=300 in a four-sided square cavity (Fig. 1(b)). It is observed that the limiting value of Re for four-sided cavity is very close to previously reported value in [5], however for motion of non-facing walls in two-sided cavity, multiple solutions are seen to exist even for Re=975, which is significantly lower than the previously reported threshold value of 1071 [5].

References

[1]     Perumal, D. A., & Dass, A. K. (2011). Multiplicity of steady solutions in two-dimensional lid-driven cavity flows by Lattice Boltzmann Method. Computers & Mathematics with Applications, 61(12), 3711-3721.

[2]     Albensoeder, S., Kuhlmann, H. C., & Rath, H. J. (2001). Multiplicity of steady two-dimensional flows in two-sided lid-driven cavities. Theoretical and Computational Fluid Dynamics, 14(4), 223-241.

[3]     Kuhlmann, H. C., Wanschura, M., & Rath, H. J. (1997). Flow in two-sided lid-driven cavities: non-uniqueness, instabilities, and cellular structures. Journal of Fluid Mechanics, 336, 267-299.

[4]     Lemée, T., Kasperski, G., Labrosse, G., & Narayanan, R. (2015). Multiple stable solutions in the 2D symmetrical two-sided square lid-driven cavity. Computers & Fluids, 119, 204-212.

[5]     Wahba, E. M. (2009). Multiplicity of states for two-sided and four-sided lid driven cavity flows. Computers & Fluids, 38(2), 247-253.

[6]     Blohm, C. H., & Kuhlmann, H. C. (2002). The two-sided lid-driven cavity: experiments on stationary and time-dependent flows. Journal of Fluid Mechanics, 450, 67-95.

[7]     Luo, W. J., & Yang, R. J. (2007). Multiple fluid flow and heat transfer solutions in a two-sided lid-driven cavity. International journal of heat and mass transfer, 50(11), 2394-2405.

[8]     Kalita, J. C., Dalal, D. C., & Dass, A. K. (2002). A class of higher order compact schemes for the unsteady two-dimensional convection–diffusion equation with variable convection coefficients. International Journal for Numerical Methods in Fluids, 38(12), 1111-1131.

[9]     Prasad, C., & Dass, A. K. (2016). Use of an HOC scheme to determine the existence of multiple steady states in the antiparallel lid driven flow in a two-sided square cavity. International Journal for Numerical Methods in Fluids, submitted for publication.