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In this paper, bifurcations in the solution of the Navier–Stokes equations are studied and multiple solutions of the driven semi-elliptical cavity flow are presented. The two-dimensional steady incompressible driven viscous flow in a semi-elliptical cavity is solved numerically. To this end, the problem is formulated using an elliptic coordinate system that transforms the geometry conformally and provides a body fitted coordinate system. The presented results show that above a bifurcation Reynolds number the solution of the governing flow equations bifurcates and there exist multiple solutions for a particular Reynolds number when the aspect ratio of the semi-elliptical cavity geometry is 0.26 ⩽D⩽ 0.8. The bifurcation Reynolds numbers for different aspect ratios and also multiple solutions at different Reynolds numbers are presented in detail. [ABSTRACT FROM AUTHOR]

In this study, the numerical performance of the fourth-order compact formulation of the steady 2-D incompressible Navier–Stokes (NS) equations introduced by Erturk et al. (Int. J. Numer. Methods Fluids 2006; 50:421–436) will be presented. The benchmark-driven cavity flow problem will be solved using the introduced compact fourth-order formulation of the NS equations with two different line iterative semi-implicit methods for both second- and fourth-order spatial accuracy. The extra CPU work needed for increasing the spatial accuracy from second-order (𝒪(Δx2)) formulation to fourth-order (𝒪(Δx4)) formulation will be presented. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]